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 non zero integer. That is, if for a prime q there is no smaller prime p and nonzero integer b such that q = p + 2b2, 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, 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(12), or 131 + 2(22), etc.
In fact, many primes have more than one such representation. Given a twin prime, the larger prime of the pair has a Goldbach representation (namely, a representation as the sum of two primes) of p + 2(12). If that prime is the largest of a prime quadruplet, p + 8, then p + 2(22) is also valid. Sloane's lists odd numbers with at least n Goldbach representations. Leonhard Euler observed that as numbers get larger, they have more representations of the form
p+2b2
There also exist odd composite Stern numbers: the only known ones are 5777 and 5993. Goldbach once incorrectly conjectured that all Stern numbers are prime. (See for odd Stern numbers)
Christian Goldbach conjectured in a letter to Leonhard Euler that every odd integer is of the form p + 2b2 for integer b and prime p. Laurent Hodges believes that Stern became interested in the problem after reading a book of Goldbach's correspondence. At the time, 1 was considered a prime, so 3 was not considered a Stern prime given the representation 1 + 2(12). The rest of the list remains the same under either definition.