In
mathematics, a Newman–Shanks–Williams prime (NSW prime) is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' which can be written in the form
:
NSW primes were first described by
Morris Newman,
Daniel Shanks
Daniel Shanks (January 17, 1917 – September 6, 1996) was an American mathematician who worked primarily in numerical analysis and number theory. He was the first person to compute π to 100,000 decimal places.
Life and education
Shanks was b ...
and
Hugh C. Williams in 1981 during the study of
finite simple group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s with square order.
The first few NSW primes are
7,
41,
239
__NOTOC__
Year 239 (Roman numerals, 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 frequentl ...
, 9369319, 63018038201, … , corresponding to the indices 3, 5, 7, 19, 29, … .
The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
''S'' alluded to in the formula can be described by the following
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
:
:
:
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
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
convergents to
.
Further reading
*
External links
The Prime Glossary: NSW number
{{DEFAULTSORT:Newman-Shanks-Williams prime
Classes of prime numbers
Unsolved problems in mathematics