Newman-Shanks-Williams prime

From Wikipedia, the free encyclopedia

This can be abbreviated to NSW, which is also the abbreviation of the state of New South Wales in Australia.

In mathematics, a Newman-Shanks-Williams prime (often abbreviated NSW prime) is a certain kind of prime number. A prime p is an NSW prime if it is a Newman-Shanks-Williams number; that is, if it can be written in the form

$S_{2m+1}=\frac{(1+\sqrt{2})^{2m+1}+(1-\sqrt{2})^{2m+1}}{2}$

NSW primes were first described by M. Newman, D. Shanks and H. C. Williams in 1981 during the study of finite groups with square order.

The first few NSW primes are 7, 41, 239, 9369319, 63018038201, ... (sequence A088165 in OEIS), corresponding to the indices 3, 5, 7, 19, 29, ... (sequence A005850 in OEIS).

The sequence $S$ alluded to in the formula can be described by the following recurrence relation:

S 0 = 1

S 1 = 1

$S_n=2S_{n-1}+S_{n-2}\qquad\mbox{for all }n\geq2.$ .

The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, ... (sequence A001333 in OEIS). These numbers also appear in the continued fraction convergents to √2.