A Stern prime, named for

Moritz Abraham Stern Moritz Abraham Stern (29 June 1807 – 30 January 1894) was a German mathematician. Stern became ''Ordinarius'' (full professor) at Göttingen University in 1858, succeeding Carl Friedrich Gauss. Stern was the first Jewish full professor at a Germ ...

, is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that is not the sum of a smaller prime and twice the

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

of a non zero

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. That is, if for a prime ''q'' there is no smaller prime ''p'' and nonzero integer ''b'' such that ''q'' = ''p'' + 2''b''², then ''q'' is a Stern prime. The known Stern primes are : 2, 3, 17,

137 137 may refer to: *137 (number) *137 BC *AD 137 *137 (album), an album by The Pineapple Thief *137 (MBTA bus) The Massachusetts Bay Transportation Authority bus division operates bus routes in the Boston, Massachusetts metropolitan area. All ro ...

, 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 A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...

, 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 In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prim ...

, ''p'' + 8, then ''p'' + 2(2²) is also valid. Sloane's lists odd numbers with at least ''n'' Goldbach representations.

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

observed that as numbers get larger, they have more representations of the form

p + 2b^2

, suggesting that there may be a largest number with no such representations; i.e., the above list of Stern primes might be not only finite, but 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. 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 Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling ...

conjectured in a letter to Leonhard Euler that every odd integer is of the form ''p'' + 2''b''² 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

* {{Prime number classes Classes of prime numbers