Solved problem 58.
parent
0ee19fa69f
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2aac4a6d4a
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@ -17,19 +17,250 @@
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"\n",
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"Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.\n",
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"\n",
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"37 36 35 34 33 32 31\n",
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"38 17 16 15 14 13 30\n",
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"39 18 5 4 3 12 29\n",
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"40 19 6 1 2 11 28\n",
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"41 20 7 8 9 10 27\n",
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"42 21 22 23 24 25 26\n",
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"43 44 45 46 47 48 49\n",
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" 37 36 35 34 33 32 31\n",
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" 38 17 16 15 14 13 30\n",
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" 39 18 5 4 3 12 29\n",
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" 40 19 6 1 2 11 28\n",
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" 41 20 7 8 9 10 27\n",
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" 42 21 22 23 24 25 26\n",
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" 43 44 45 46 47 48 49\n",
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"\n",
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"It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.\n",
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"\n",
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"If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"def get_last_corner_value(side_length):\n",
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" return side_length * side_length\n",
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"\n",
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"assert(get_last_corner_value(1) == 1)\n",
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"assert(get_last_corner_value(3) == 9)\n",
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"assert(get_last_corner_value(5) == 25)\n",
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"\n",
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"\n",
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"def get_corner_values(side_length):\n",
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" if side_length == 1:\n",
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" return [1]\n",
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" return [get_last_corner_value(side_length) - i * (side_length - 1)\n",
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" for i in range(0, 4)][::-1]\n",
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"\n",
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"assert(get_corner_values(1) == [1])\n",
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"assert(get_corner_values(3) == [3, 5, 7, 9])\n",
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"assert(get_corner_values(5) == [13, 17, 21, 25])\n",
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"assert(get_corner_values(7) == [31, 37, 43, 49])\n",
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"\n",
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"\n",
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"def get_diagonal_values(side_length):\n",
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" return [corner_value\n",
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" for length in range(1, side_length + 1, 2)\n",
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" for corner_value in get_corner_values(length)\n",
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" ]\n",
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"\n",
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"assert(get_diagonal_values(1) == [1])\n",
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"assert(get_diagonal_values(3) == [1, 3, 5, 7, 9])\n",
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"assert(get_diagonal_values(5) == [1, 3, 5, 7, 9, 13, 17, 21, 25])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def expmod(base, exp, m):\n",
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" if exp == 0:\n",
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" return 1\n",
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" if (exp % 2 == 0):\n",
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" return (expmod(base, exp // 2, m) ** 2 % m)\n",
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" return (base * expmod(base, exp - 1, m) % m)\n",
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" \n",
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"\n",
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"def fermat_test(n):\n",
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" if n < 40:\n",
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" return n in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]\n",
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" a = n - 3\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 5\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 7\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 11\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 13\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 17\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 19\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 23\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 29\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 31\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 37\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" a = n - 39\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" return True\n",
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"\n",
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"def trial_division(n):\n",
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" a = [] \n",
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" if n % 2 == 0:\n",
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" a.append(2)\n",
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" while n % 2 == 0:\n",
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" n //= 2\n",
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" f = 3\n",
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" while f * f <= n:\n",
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" if n % f == 0:\n",
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" a.append(f)\n",
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" while n % f == 0:\n",
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" n //= f\n",
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" else:\n",
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" f += 2 \n",
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" if n != 1:\n",
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" a.append(n)\n",
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" return a\n",
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"\n",
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"def is_prime(n):\n",
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" if n == 1:\n",
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" return False\n",
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" return trial_division(n)[0] == n\n",
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"\n",
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"assert(fermat_test(3))\n",
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"assert(fermat_test(107))\n",
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"assert(fermat_test(108) == False)\n",
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"assert(fermat_test(109))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"n: 26241 count_total: 52481 count_primes: 5248 ratio: 0.09999809454850327\n"
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]
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}
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],
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"source": [
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"def get_solution():\n",
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" n = 1\n",
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" count_primes = 0\n",
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" count_total = 0\n",
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" while True:\n",
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" for v in get_corner_values(n):\n",
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" count_total += 1\n",
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" if is_prime(v):\n",
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" count_primes += 1\n",
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" ratio = count_primes / count_total\n",
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" if ratio != 0 and ratio < 0.10:\n",
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" print(\"n: {} count_total: {} count_primes: {} ratio: {}\".format(n, count_total, count_primes, ratio))\n",
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" return n\n",
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" n += 2\n",
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"\n",
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"s = get_solution()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"26241\n"
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]
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}
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],
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"source": [
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"print(s)\n",
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"assert(s == 26241)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"I actually got different results for the Fermat test and for the prime test which relies on reliable computation. I will actually try to solve the problem with ramdonized Fermat test now."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"import random\n",
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"\n",
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"def is_prime(n):\n",
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" if n == 1:\n",
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" return False\n",
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" for _ in range(100):\n",
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" a = n - random.randint(1, n - 1)\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" return True\n",
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"\n",
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"assert(fermat_test(3))\n",
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"assert(fermat_test(107))\n",
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"assert(fermat_test(108) == False)\n",
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"assert(fermat_test(109))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913\n"
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]
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}
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],
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"source": [
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"s = get_solution()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Something seems to be off with my Fermat test... \n",
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"\n",
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"Seems like there are systematic errors with the Fermat tests. Certain primes cannot be deteced.\n",
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"\n",
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"Try this algorithm instead [https://en.wikipedia.org/wiki/Miller–Rabin_primality_test](https://en.wikipedia.org/wiki/Miller–Rabin_primality_test)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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@ -41,7 +272,7 @@
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}
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],
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"metadata": {
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"completion_date": "",
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"completion_date": "Sat, 29 Dec 2018, 09:00",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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@ -59,7 +290,13 @@
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"pygments_lexer": "ipython3",
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"version": "3.6.5"
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},
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"tags": []
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"tags": [
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"prime",
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"fermat",
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"spiral",
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"diagonal",
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"todo"
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]
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},
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"nbformat": 4,
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"nbformat_minor": 2
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@ -0,0 +1,73 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Cyclical figurate numbers (Euler Problem 61)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"[https://projecteuler.net/problem=61](https://projecteuler.net/problem=61)\n",
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"\n",
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"Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:\n",
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"\n",
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"Triangle\t \tP3,n=n(n+1)/2\t \t1, 3, 6, 10, 15, ...\n",
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"\n",
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"Square\t \tP4,n=n2\t \t1, 4, 9, 16, 25, ...\n",
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"\n",
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"Pentagonal\t \tP5,n=n(3n−1)/2\t \t1, 5, 12, 22, 35, ...\n",
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"\n",
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"Hexagonal\t \tP6,n=n(2n−1)\t \t1, 6, 15, 28, 45, ...\n",
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"\n",
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"Heptagonal\t \tP7,n=n(5n−3)/2\t \t1, 7, 18, 34, 55, ...\n",
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"\n",
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"Octagonal\t \tP8,n=n(3n−2)\t \t1, 8, 21, 40, 65, ...\n",
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"\n",
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"The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.\n",
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"\n",
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"The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).\n",
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"Each polygonal type: triangle (P3,127=8128), square (P4,91=8281), and pentagonal (P5,44=2882), is represented by a different number in the set.\n",
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"This is the only set of 4-digit numbers with this property.\n",
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"Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.5"
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},
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"tags": []
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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@ -0,0 +1,56 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Cubic permutations (Euler Problem 62)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"[https://projecteuler.net/problem=62](https://projecteuler.net/problem=62)\n",
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"\n",
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"The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.\n",
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"\n",
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"Find the smallest cube for which exactly five permutations of its digits are cube."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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||||
"pygments_lexer": "ipython3",
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||||
"version": "3.6.5"
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},
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"tags": []
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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@ -0,0 +1,54 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Powerful digit counts (Euler Problem 63)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power.\n",
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"\n",
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"How many n-digit positive integers exist which are also an nth power?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
|
||||
"collapsed": true
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||||
},
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||||
"outputs": [],
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"source": []
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}
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],
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||||
"metadata": {
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"completion_date": "",
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"kernelspec": {
|
||||
"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
|
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
|
||||
"file_extension": ".py",
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||||
"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.6.5"
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||||
},
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||||
"tags": []
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},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 2
|
||||
}
|
||||
|
|
@ -11780,13 +11780,15 @@ div#notebook {
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p><a href="https://projecteuler.net/problem=58">https://projecteuler.net/problem=58</a></p>
|
||||
<p>Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.</p>
|
||||
<p>37 36 35 34 33 32 31
|
||||
<pre><code>37 36 35 34 33 32 31
|
||||
38 17 16 15 14 13 30
|
||||
39 18 5 4 3 12 29
|
||||
40 19 6 1 2 11 28
|
||||
41 20 7 8 9 10 27
|
||||
42 21 22 23 24 25 26
|
||||
43 44 45 46 47 48 49</p>
|
||||
43 44 45 46 47 48 49
|
||||
|
||||
</code></pre>
|
||||
<p>It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.</p>
|
||||
<p>If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?</p>
|
||||
</div>
|
||||
|
|
@ -11794,6 +11796,256 @@ div#notebook {
|
|||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_last_corner_value</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="n">side_length</span> <span class="o">*</span> <span class="n">side_length</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="mi">9</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="mi">25</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_corner_values</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">side_length</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="n">side_length</span><span class="p">)</span> <span class="o">-</span> <span class="n">i</span> <span class="o">*</span> <span class="p">(</span><span class="n">side_length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="mi">25</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">31</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">43</span><span class="p">,</span> <span class="mi">49</span><span class="p">])</span>
|
||||
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_diagonal_values</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="n">corner_value</span>
|
||||
<span class="k">for</span> <span class="n">length</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">side_length</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">corner_value</span> <span class="ow">in</span> <span class="n">get_corner_values</span><span class="p">(</span><span class="n">length</span><span class="p">)</span>
|
||||
<span class="p">]</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="mi">25</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="mi">1</span>
|
||||
<span class="k">if</span> <span class="p">(</span><span class="n">exp</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="p">(</span><span class="n">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="p">(</span><span class="n">base</span> <span class="o">*</span> <span class="n">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="k">def</span> <span class="nf">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o"><</span> <span class="mi">40</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">39</span><span class="p">]</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">5</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">7</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">11</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">13</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">17</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">19</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">23</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">29</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">31</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">37</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">39</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">trial_division</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
|
||||
<span class="n">f</span> <span class="o">=</span> <span class="mi">3</span>
|
||||
<span class="k">while</span> <span class="n">f</span> <span class="o">*</span> <span class="n">f</span> <span class="o"><=</span> <span class="n">n</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
|
||||
<span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">n</span> <span class="o">//=</span> <span class="n">f</span>
|
||||
<span class="k">else</span><span class="p">:</span>
|
||||
<span class="n">f</span> <span class="o">+=</span> <span class="mi">2</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">a</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="n">trial_division</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">n</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">107</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">108</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">109</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_solution</span><span class="p">():</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">count_primes</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="n">count_total</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
|
||||
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">get_corner_values</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">count_total</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
<span class="k">if</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
|
||||
<span class="n">count_primes</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
<span class="n">ratio</span> <span class="o">=</span> <span class="n">count_primes</span> <span class="o">/</span> <span class="n">count_total</span>
|
||||
<span class="k">if</span> <span class="n">ratio</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">ratio</span> <span class="o"><</span> <span class="mf">0.10</span><span class="p">:</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"n: </span><span class="si">{}</span><span class="s2"> count_total: </span><span class="si">{}</span><span class="s2"> count_primes: </span><span class="si">{}</span><span class="s2"> ratio: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">count_total</span><span class="p">,</span> <span class="n">count_primes</span><span class="p">,</span> <span class="n">ratio</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="n">n</span>
|
||||
<span class="n">n</span> <span class="o">+=</span> <span class="mi">2</span>
|
||||
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">get_solution</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
<div class="output_area">
|
||||
<div class="prompt"></div>
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>n: 26241 count_total: 52481 count_primes: 5248 ratio: 0.09999809454850327
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">26241</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
<div class="output_area">
|
||||
<div class="prompt"></div>
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>26241
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I actually got different results for the Fermat test and for the prime test which relies on reliable computation. I will actually try to solve the problem with ramdonized Fermat test now.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">random</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">107</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">108</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">109</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="n">get_solution</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
<div class="output_area">
|
||||
<div class="prompt"></div>
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Something seems to be off with my Fermat test...</p>
|
||||
<p>Seems like there are systematic errors with the Fermat tests. Certain primes cannot be deteced.</p>
|
||||
<p>Try this algorithm instead <a href="https://en.wikipedia.org/wiki/Miller–Rabin_primality_test">https://en.wikipedia.org/wiki/Miller–Rabin_primality_test</a>.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [ ]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
|
|
|
|||
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
|
|
@ -917,12 +917,22 @@
|
|||
</tr>
|
||||
|
||||
|
||||
<tr class="table-warning">
|
||||
<tr>
|
||||
|
||||
<td><a href="EulerProblem058.html">Problem 058</a></td>
|
||||
<td></td>
|
||||
<td>Sat, 29 Dec 2018, 09:00</td>
|
||||
<td>
|
||||
|
||||
<kbd>prime</kbd>
|
||||
|
||||
<kbd>fermat</kbd>
|
||||
|
||||
<kbd>spiral</kbd>
|
||||
|
||||
<kbd>diagonal</kbd>
|
||||
|
||||
<kbd>todo</kbd>
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
|
||||
|
|
@ -947,6 +957,36 @@
|
|||
</tr>
|
||||
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem061.html">Problem 061</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem062.html">Problem 062</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem063.html">Problem 063</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
|
||||
<td><a href="EulerProblem067.html">Problem 067</a></td>
|
||||
|
|
|
|||
Loading…
Reference in New Issue