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| In [[mathematics]], a '''Newman–Shanks–Williams prime (NSW prime)''' is a [[prime number]] ''p'' which can be written in the form
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| :<math>S_{2m+1}=\frac{\left(1 + \sqrt{2}\right)^{2m+1} + \left(1 - \sqrt{2}\right)^{2m+1}}{2}.</math>
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| NSW primes were first described by [[Morris Newman]], [[Daniel Shanks]] and [[Hugh C. Williams]] in 1981 during the study of [[finite simple group]]s with square order.
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| The first few NSW primes are [[7 (number)|7]], [[41 (number)|41]], [[239 (number)|239]], 9369319, 63018038201, … {{OEIS|id=A088165}}, corresponding to the indices 3, 5, 7, 19, 29, … {{OEIS|id=A005850}}.
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| The [[sequence]] ''S'' alluded to in the formula can be described by the following [[recurrence relation]]:
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| :<math>S_0=1 \, </math> | |
| :<math>S_1=1 \, </math>
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| :<math>S_n=2S_{n-1}+S_{n-2}\qquad\text{for all }n\geq 2.</math>
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| The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … {{OEIS|id=A001333}}. Each term in this sequence is half the corresponding term in the sequence of [[Pell number|companion Pell numbers]]. These numbers also appear in the [[continued fraction]] convergents to √2.
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| == Further reading == | |
| *{{cite journal |first=M. |last=Newman |first2=D. |last2=Shanks |lastauthoramp=yes |first3=H. C. |last3=Williams |title=Simple groups of square order and an interesting sequence of primes |journal=[[Acta Arithmetica]] |volume=38 |issue=2 |year=1980/1981 |pages=129–140 |doi= }}
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| == External links ==
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| * [http://primes.utm.edu/glossary/page.php?sort=NSWNumber The Prime Glossary: NSW number]
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| {{Prime number classes}}
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| {{DEFAULTSORT:Newman-Shanks-Williams prime}}
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| [[Category:Classes of prime numbers]]
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