Number of Newman-Shanks-Williams

In Mathematical, a prime number of Newman-Shanks-Williams (often shortened prime number NSW ) east is a Prime number particular which is defined in the following way:

A prime number p is a prime number NSW if it can be written in the form:

$S\_\; \{2m+1\}\; =\; \backslash \; frac\; \{(1+\; \backslash \; sqrt\; \{2\})\; ^\; \{2m+1\}\; +\; (1\; \backslash \; sqrt\; \{2\})\; ^\; \{2m+1\}\}\; \{2\}$

Prime numbers NSW were described in first by Mr. Newman, D. Shanks and H.C. Williams in 1981 during the study of the finished groups of a square nature.

First small numbers first NSW are 7, 41, 239, 9369319,63018038201,… electronic Encyclopédie of the whole continuations (id=A088165), corresponding to indices 3,5,7,19,29,… electronic Encyclopédie of the whole continuations (id=A005850).

The continuation $S\; \backslash ,$ allocated with the formula can be described by the relation of following recurrence:

$S\_0=1\; \backslash ,$

$S\_1=1\; \backslash ,$

$S\_n=2S\_\; \{n-1\}\; +S\_\; \{N2\}\; \backslash \; qquad\; \backslash \; mbox\; \{for\; all\}\; N\; \backslash \; geq2\; \backslash ,$.

The first minor terms of the continuation are 1,1,3,7,17,41,99,… (continuation on A001333). These numbers also appear in the continued Fraction convergent of $\backslash \; sqrt\; 2\; \backslash ,$.

External bond

• the glossary of the prime numbers: number NSW (in English)

To go further

• Mr. Newman, D. Shanks and H.C. Williams, Simple groups off public garden order and year interesting sequence off premiums , Acta. Arith., 38:2 (1980/81) 129-140.

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