# Definition:Stern Number

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## Definition

A **Stern number** is an odd 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 numbers begins:

- $1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993$

It is not known whether there are any more.

## Also see

- Results about
**Stern numbers**can be found here.

## 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 integers that cannot be represented in the form $2 a^2 + p$ where $a > 0$.

The odd integers that he and his students found were named Stern numbers 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