Newman-Shanks-Williams prime

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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 prime number p which 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, … (A005850).

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.