Problem 46

Goldbach’s other conjecture

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

$9 = 7 + 2\times 1^2$

$15 = 7 + 2\times 2^2$

$21 = 3 + 2\times 3^2$

$25 = 7 + 2\times 3^2$

$27 = 19 + 2\times 2^2$

$33 = 31 + 2\times 1^2$

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?

克里斯蒂安·哥德巴赫曾经猜想，每个奇合数都可以写成一个素数和一个平方的两倍之和。

$9 = 7 + 2\times 1^2$

$15 = 7 + 2\times 2^2$

$21 = 3 + 2\times 3^2$

$25 = 7 + 2\times 3^2$

$27 = 19 + 2\times 2^2$

$33 = 31 + 2\times 1^2$

最终这个猜想被推翻了。

不能写成一个素数和一个平方的两倍之和的最小奇合数是多少？