https://projecteuler.net/problem=58
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
-37 36 35 34 33 32 31 +
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
+43 44 45 46 47 48 49
+
+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%.
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%?
def get_last_corner_value(side_length):
+ return side_length * side_length
+
+assert(get_last_corner_value(1) == 1)
+assert(get_last_corner_value(3) == 9)
+assert(get_last_corner_value(5) == 25)
+
+
+def get_corner_values(side_length):
+ if side_length == 1:
+ return [1]
+ return [get_last_corner_value(side_length) - i * (side_length - 1)
+ for i in range(0, 4)][::-1]
+
+assert(get_corner_values(1) == [1])
+assert(get_corner_values(3) == [3, 5, 7, 9])
+assert(get_corner_values(5) == [13, 17, 21, 25])
+assert(get_corner_values(7) == [31, 37, 43, 49])
+
+
+def get_diagonal_values(side_length):
+ return [corner_value
+ for length in range(1, side_length + 1, 2)
+ for corner_value in get_corner_values(length)
+ ]
+
+assert(get_diagonal_values(1) == [1])
+assert(get_diagonal_values(3) == [1, 3, 5, 7, 9])
+assert(get_diagonal_values(5) == [1, 3, 5, 7, 9, 13, 17, 21, 25])
+def expmod(base, exp, m):
+ if exp == 0:
+ return 1
+ if (exp % 2 == 0):
+ return (expmod(base, exp // 2, m) ** 2 % m)
+ return (base * expmod(base, exp - 1, m) % m)
+
+
+def fermat_test(n):
+ if n < 40:
+ return n in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]
+ a = n - 3
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 5
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 7
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 11
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 13
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 17
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 19
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 23
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 29
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 31
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 37
+ if not expmod(a, n, n) == a:
+ return False
+ a = n - 39
+ if not expmod(a, n, n) == a:
+ return False
+ return True
+
+def trial_division(n):
+ a = []
+ if n % 2 == 0:
+ a.append(2)
+ while n % 2 == 0:
+ n //= 2
+ f = 3
+ while f * f <= n:
+ if n % f == 0:
+ a.append(f)
+ while n % f == 0:
+ n //= f
+ else:
+ f += 2
+ if n != 1:
+ a.append(n)
+ return a
+
+def is_prime(n):
+ if n == 1:
+ return False
+ return trial_division(n)[0] == n
+
+assert(fermat_test(3))
+assert(fermat_test(107))
+assert(fermat_test(108) == False)
+assert(fermat_test(109))
+def get_solution():
+ n = 1
+ count_primes = 0
+ count_total = 0
+ while True:
+ for v in get_corner_values(n):
+ count_total += 1
+ if is_prime(v):
+ count_primes += 1
+ ratio = count_primes / count_total
+ if ratio != 0 and ratio < 0.10:
+ print("n: {} count_total: {} count_primes: {} ratio: {}".format(n, count_total, count_primes, ratio))
+ return n
+ n += 2
+
+s = get_solution()
+print(s)
+assert(s == 26241)
+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.
+import random
+
+def is_prime(n):
+ if n == 1:
+ return False
+ for _ in range(100):
+ a = n - random.randint(1, n - 1)
+ if not expmod(a, n, n) == a:
+ return False
+ return True
+
+assert(fermat_test(3))
+assert(fermat_test(107))
+assert(fermat_test(108) == False)
+assert(fermat_test(109))
+s = get_solution()
+Something seems to be off with my Fermat test...
+Seems like there are systematic errors with the Fermat tests. Certain primes cannot be deteced.
+Try this algorithm instead https://en.wikipedia.org/wiki/Miller–Rabin_primality_test.
+https://projecteuler.net/problem=61
+Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
+Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ...
+Square P4,n=n2 1, 4, 9, 16, 25, ...
+Pentagonal P5,n=n(3n−1)/2 1, 5, 12, 22, 35, ...
+Hexagonal P6,n=n(2n−1) 1, 6, 15, 28, 45, ...
+Heptagonal P7,n=n(5n−3)/2 1, 7, 18, 34, 55, ...
+Octagonal P8,n=n(3n−2) 1, 8, 21, 40, 65, ...
+The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.
+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). +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. +This is the only set of 4-digit numbers with this property. +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.
+
+https://projecteuler.net/problem=62
+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.
+Find the smallest cube for which exactly five permutations of its digits are cube.
+
+The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power.
+How many n-digit positive integers exist which are also an nth power?
+
+