Stern prime

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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 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 (sequence A042978 in OEIS).

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 is 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 representation of this sort. Given a twin prime, the larger prime of the pair has, if nothing else, a Goldbach representation of p + 2(1²). And if that prime is the largest of a prime quadruplet, p + 8, then p + 2(2²) is also available. Sloane's OEISA007697 lists odd numbers with at least n Goldbach representations. Leonhard Euler observed that as the numbers get larger, they get more representations of the form p + 2b^2, suggesting that there might be a largest number with zero such 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 their Goldbach representations would suggest.

Christian Goldbach conjectured in a letter to Leonhard Euler that every odd integer is of the form p + 2b² with b allowed to be any integer, including zero. Laurent Hodges believes that 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(1²). The rest of the list remains the same.

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