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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?

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From wikipedia: "The number one is a unit; it is neither prime nor composite" –  P.. Jun 5 '13 at 6:27
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2 Answers 2

up vote 2 down vote accepted

If we take primes to be positive, it cannot be done. This is because primes are greater than $1$, and twice a square number is positive.

This is not contradictory to the problem though, because $1$ is not composite.

If we allow primes to be negative (considering prime elements in the ring of integers), we can write $1=-7+2\cdot2^2$.

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Thanks, didn't see the word composite. –  moray95 Jun 5 '13 at 6:14
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$1$ is not the product of two smaller positive integers. So it isn't composite, and the conjecture doesn't apply.

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