In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Morris may refer to:
Places
Australia
*St Morris, South Australia, place in South Australia
Canada
* Morris Township, Ontario, now part of the municipality of Morris-Turnberry
* Rural Municipality of Morris, Manitoba
** Morris, Manit ...
,
Daniel Shanks and
Hugh C. Williams
Hugh Cowie Williams (born 23 July 1943) is a Canadian mathematician. He deals with number theory and cryptography.
Early life
Williams studied mathematics at the University of Waterloo (bachelor's degree 1966, master's degree 1967), where he ...
in 1981 during the study of
finite simple groups with square order.
The first few NSW primes are
7,
41,
239
__NOTOC__
Year 239 ( CCXXXIX) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gordianus and Aviola (or, less frequently, year 992 ''A ...
, 9369319, 63018038201, … , corresponding to the indices 3, 5, 7, 19, 29, … .
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, … . 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
.
Further reading
*
External links
The Prime Glossary: NSW number
{{DEFAULTSORT:Newman-Shanks-Williams prime
Classes of prime numbers
Unsolved problems in mathematics