Stern prime
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
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(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 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
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
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.
References
- Laurent Hodges, A lesser-known Goldbach conjecture