Definition:Stern Prime
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
- 1993: Laurent Hodges: A Lesser-Known Goldbach Conjecture (Math. Mag. Vol. 66: pp. 45 – 47) www.jstor.org/stable/2690477