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.
\begin{align}
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
\end{align}
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?

Let’s test odd composite numbers one by one.

Note: An odd composite number is a composite number that is odd.

Composite numbers have more than two factors (unlike prime numbers), and since our number needs to be odd, even numbers are immediately ruled out.

Let’s go through the odd composite numbers from smallest to largest:

– 9 — can be written as 7 + 2*(1^2).
– 15 — can be written as 7 + 2*(2^2).
– 21 — can be written as 3 + 2*(3^2).
– 25 — can be written as 7 + 2*(3^2).
– 27 — can be written as 19 + 2*(2^2).
– 33 — can be written as 31 + 2*(1^2).
– 35 can be written as 3 + 2*(4^2).
– 39 can be written as 7 + 2*(4^2).
– 45 can be written as 7 + 2*(5^2).
– 49 can be written as 5 + 2*(6^2).
– 51 can be written as 1 + 2*(7^2).
– 55 can be written as 13 + 2*(6^2).
– 57 can be written as 41 + 2*(4^2).

Now, let’s check the number 58:
– is it composite? Yes, because 58 = 2 * 29.
– is it odd? Yes, because 58 is not divisible by 2.

But, unfortunately, no prime number exists such that when it is added to twice the square of another number, you get 58.

So, the smallest odd composite number that cannot be written as the sum of a prime and twice a square is 58.

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