Cunningham number

In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.

Definition

Cunningham numbers are a simple type of binomial number – they are of the form

b^{n}\pm 1

where b and n are integers and b is not a perfect power. They are denoted C^±(b, n).

Terms

The first fifteen terms in the sequence of Cunningham numbers are:

3, 5, 7, 8, 9, 10, 15, 17, 24, 26, 28, 31, 33, 35, 37, ... (sequence A080262 in the OEIS)

Properties

There are infinitely many even and odd Cunningham numbers. Provable through observation that the infinite series

6^{n}\pm 1

5^{n}\pm 1

are both contained within the Cunningham numbers, and contain only Odd and Even numbers respectively.

- By the same logic, there are Infinitely many Cunningham numbers which are 7 modulo 10, and same for 6 modulo 10.

Primality

Establishing whether or not a given Cunningham number is prime has been the main focus of research around this type of number.^[1] Two particularly famous families of Cunningham numbers in this respect are the Fermat numbers, which are those of the form C⁺(2, 2^m), and the Mersenne numbers, which are of the form C⁻(2, n).

Cunningham worked on gathering together all known data on which of these numbers were prime. In 1925 he published tables which summarised his findings with H. J. Woodall, and much computation has been done in the intervening time to fill these tables.^[2]

References

^ J. Brillhart, D. H. Lehmer, J. Selfridge, B. Tuckerman, and S. S. Wagstaff Jr., Factorizations of bⁿ±1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n), 3rd ed. Providence, RI: Amer. Math. Soc., 1988.
^ R. P. Brent and H. J. J. te Riele, Factorizations of aⁿ±1, 13≤a<100 Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.

External links

[1] J. Brillhart, D. H. Lehmer, J. Selfridge, B. Tuckerman, and S. S. Wagstaff Jr., Factorizations of bⁿ±1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n), 3rd ed. Providence, RI: Amer. Math. Soc., 1988.

[2] R. P. Brent and H. J. J. te Riele, Factorizations of aⁿ±1, 13≤a<100 Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.

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