From Problem 46 of Project Euler :
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
$$9 = 7 + 2 \cdot 1^2$$
$$15 = 7 + 2 \cdot 2^2$$
$$21 = 3 + 2 \cdot 3^2$$
$$25 = 7 + 2 \cdot 3^2$$
$$27 = 19 + 2 \cdot 2^2$$
$$33 = 31 + 2 \cdot 1^2$$
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?
As soon as I read this problem, I've immediately fought that the smallest odd number 1 was not on the list of examples. Then I've thought that 1 can be the answer. The only way that I could write 1 close to the problem $1=1+2 \cdot 0^2$, Which is of course false. I've also tried to give 1 as answer, but the site didn't accepted. So how can we write 1 as a sum of a prime and twice a square?