Smallest Odd Number not of form 2 a squared plus p
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
- 1856: Moritz A. Stern: Sur un assertion de Goldbach relative aux nombres impairs (Nouv. Ann. Math. Vol. 15: pp. 23 – 24)
- 1993: Laurent Hodges: A Lesser-Known Goldbach Conjecture (Math. Mag. Vol. 66: pp. 45 – 47) www.jstor.org/stable/2690477
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$
- except that in his statement of the problem, he omits to mention $1$ and $3$.