# NSW number

NSW stands for Newman, Shanks, and Williams who wrote a paper [NSW1981] in the 1970’s on the integers of the formSThis sequence begins: S_{2m+1}= ((1 + sqrt(2))^{2m+1}+ (1 - sqrt(2))^{2m+1})/2.

_{1}=1, S

_{3}=7, S

_{5}=41, S

_{7}=239, and S

_{9}=1393. (These numbers arise when addressing the question "is there a finite simple group whose order is a square?")

The **NSW primes** are obviously prime NSW numbers. The first few
are S_{p} where *p* = 3, 5, 7, 19, 29, 47, 59,
163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, and 9679.
(The next is most likely 28753, a probable-prime.)

**See Also:** FibonacciPrime

**Related pages** (outside of this work)

- A002315 Sloane's Sequence

**References:**

- BBLR1998
E. Barcucci,S. Brunetti,A. Del LungoandF. Del Ristoro, "A combinatorial interpretation of the recurrencef_{n+1}=6f_{n}-f_{n-1},"Discrete Math.,190(1998) 235--240.MR 99f:05001- NSW1980
M. Newman,D. ShanksandH. C. Williams, "Simple groups of square order and an interesting sequence of primes.,"Acta. Arith.,38:2 (1980/81) 129--140.MR 82b:20022- Ribenboim95 (p. 367-369)
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, 1995. New York, NY, pp. xxiv+541, ISBN 0-387-94457-5.MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]

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