Definition:Stern Prime

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Definition

A Stern prime is a prime number which can not be represented in the form:

$2 a^2 + p$

where:

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


Sequence

The sequence of Stern primes begins:

$2, 3, 17, 137, 227, 977, 1187, 1493$


It is not known whether there are any more.


Also see


Source of Name

This entry was named for Moritz Abraham Stern.


Historical Note

On reading about Goldbach's Lesser Conjecture in $1856$, 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 $a > 0$, thereby disallowing the trivial $2 \times 0^2 + p = p$.

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

The primes that he and his students found were named Stern primes by Laurent Hodges in his $1993$ paper which summarised the findings on this topic.


Sources