Smallest Odd Number not of form 2 a squared plus p

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Theorem

$17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form:

$n = 2 a^2 + p$

where:

$p$ is prime
$a \in \Z_{>0}$ is a (strictly) positive integer.


Proof

First note that $3$ is not so expressible:

\(\ds 3 - 2 \times 1^2\) \(=\) \(\ds 1\) which is not prime

and so $3$ cannot be written in such a form.


Then we have:

\(\ds 5\) \(=\) \(\ds 2 \times 1^2 + 3\)
\(\ds 7\) \(=\) \(\ds 2 \times 1^2 + 5\)
\(\ds 9\) \(=\) \(\ds 2 \times 1^2 + 7\)
\(\ds 11\) \(=\) \(\ds 2 \times 2^2 + 3\)
\(\ds 13\) \(=\) \(\ds 2 \times 1^2 + 11\)
\(\ds \) \(=\) \(\ds 2 \times 2^2 + 5\)
\(\ds 15\) \(=\) \(\ds 2 \times 1^2 + 13\)
\(\ds \) \(=\) \(\ds 2 \times 2^2 + 7\)


However, we have:

\(\ds 17 - 2 \times 1^2\) \(=\) \(\ds 15\) which is not prime
\(\ds 17 - 2 \times 2^2\) \(=\) \(\ds 9\) which is not prime

$\blacksquare$


Also see


Historical Note

The question of odd integers expressible in the form $2 a^2 + p$, for $a \ge 0$ and $p$ prime, was initially raised by Christian Goldbach in a letter to Leonhard Paul Euler dated $18$ November $1752$, in which he conjectured that all odd integers were so expressible.

At that time, $1$ was considered to be prime. Thus $1 = 2 \times 0^2 + 1$ and $3 = 2 \times 1^2 + 1$ were considered to fit the criteria, as was $17 = 0^2 + 17$.

The conjecture was believed to hold until $1856$, when Moritz Abraham Stern and his students tested all the primes to $9000$, and found the counterexamples $5777$ and $5993$.

He then went on to investigate odd numbers, and more specifically prime numbers, that cannot be represented in the form $2 a^2 + p$ where $a > 0$.

Seeming to forget about $3$, he stated that the smallest such prime number was $17$.


Sources

except that in his statement of the problem, he omits to mention $1$ and $3$.