Newman–Shanks–Williams prime

Template:Distinguish In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number which can be written in the form

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

.

NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during their study of finite simple groups with square order.Template:Sfnp

The first few NSW primes are 7, 41, 239, 9369319, 63018038201, . . . Template:OEIS, corresponding to the indices 3, 5, 7, 19, 29, . . . Template: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} = 2 S_{n - 1} + S_{n - 2} for all n \geq 2.

The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, . . . Template:OEIS. Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers. These numbers also appear in the continued fraction convergents to [[Square root of 2|Template:Radic]].