A Stern prime, named for Moritz Abraham Stern, is a prime number that is not the sum of a smaller prime and twice the square of a nonzero integer. Or, to put it algebraically, if for a prime q there is no smaller prime p and integer b <> 0 such that $q\; =\; p\; +\; 2b^2$, then q is a Stern prime. The known Stern primes are

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

So, for example,if we try subtracting from 137 the first few squares doubled in order, we get {135, 129, 119, 105, 87, 65, 39, 9}, none of which are prime. That means that 137 is a Stern prime. On the other hand, 139 is not a Stern prime, since we can express it as $137\; +\; 2(1^2)$, or $131\; +\; 2(2^2)$, etc.

In fact, many primes have more than one representation of this sort. Given a twin prime, the larger prime of the pair has, if nothing else, a Goldbach representation of $p\; +\; 2(1^2)$. And if that prime is the largest of a prime quadruplet, p + 8, then $p\; +\; 2(2^2)$ is also available. Sloane's lists odd numbers with at least n Goldbach representations.

Therefore, the above list of Stern primes might be not only finite, but also complete. According to Jud McCranie, these are the only Stern primes from among the first 100000 primes. All the known Stern primes have more efficient Waring representations than what their Goldbach representations would suggest.

Christian Goldbach conjectured in a letter to Leonhard Euler that every odd integer is of the form $p\; +\; 2b^2$ with b allowed to be any integer, including zero. Laurent Hodges believes that Moritz Stern became interested in the problem after reading a book of Goldbach's correspondence. Because in Stern's time, 1 was considered a prime, 3 was not a Stern prime because it could be represented as $1\; +\; 2(1^2)$. The rest of the list remains the same.

Reference

• Laurent Hodges, A lesser-known Goldbach conjecture

Prime numbers