Goldbach's other conjecture (Euler Problem 46)

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https://projecteuler.net/problem=46

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

$9 = 7 + 2×1^2$

$15 = 7 + 2×2^2$

$21 = 3 + 2×3^2$

$25 = 7 + 2×3^2$

$27 = 19 + 2×2^2$

$33 = 31 + 2×1^2$

It turns out that the conjecture was false.

What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

Okay, we reuse Fermat's test and brute force. Easy.

In [1]:
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):
    a = n - 3
    return expmod(a, n, n) == a
In [2]:
def twice_square(n):
    return 2 * n * n
In [18]:
n_max = 10000
twice_squares = [twice_square(n) for n in range(1, n_max + 1)]

def test_conjecture(n):
    for ts in twice_squares:
        if ts > n:
            return False
        if fermat_test(n - ts):
            return True
        
assert(test_conjecture(33))
In [19]:
for n in range(3, n_max + 1, 2):
    if not fermat_test(n) and test_conjecture(n) == False:
        s = n

print(s)
assert(s == 5777)

5777
In [ ]: