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 iff it is a Newman-Shanks-Williams number; that is, if it can be written in the form

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, ... (Sloane's A088165), corresponding to the indices 3, 5, 7, 19, 29, ... (Sloane's A005850).

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

for all .

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