Newman–Shanks–Williams prime

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

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

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

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

The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … (sequence in the 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 √2.

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