Cunningham Number

In number theory, the Cunningham number is a specific class of integers named after the English mathematician .

Content

1 Definition
2 Definition of simplicity
3 See also
4 notes
5 Links

Definition

Cunningham numbers are numbers of the form

b^{n}\pm 1

{\ displaystyle b ^ {n} \ pm 1}

where b and n are integers greater than 1, b is not an exact degree ^[1] ^[2] .

Cunningham numbers are designated ^[1] ^[2]

C^{+}(b,n)=b^{n}+1

{\ displaystyle C ^ {+} (b, n) = b ^ {n} +1}

C^{-}(b,n)=b^{n}-1.

{\ displaystyle C ^ {-} (b, n) = b ^ {n} -1.}

Definition of Simplicity

The main area of research is the search for Cunningham primes ^[1] ^[3] . The two most famous families of Cunningham numbers are Fermat numbers $C^{+}(2,2^{n})$ ${\ displaystyle C ^ {+} (2,2 ^ {n})}$ and Mersenne numbers $C^{-}(2,n).$ ${\ displaystyle C ^ {-} (2, n).}$

Allan Cunningham collected all known data on the prime numbers of this form. In 1925, tables were published that summarized the results of research by Cunningham and ; Subsequent studies have focused on filling out these tables ^[1] ^[4] .

Notes

↑ ¹ ² ³ ⁴ Weisstein, Eric W. Cunningham Number on the Wolfram MathWorld website.
↑ ¹ ² Giovanni Resta. Cunningham numbers (neopr.) . Numbers Aplenty .
↑ J. Brillhart, DH Lehmer, J. Selfridge, B. Tuckerman, and SS Wagstaff Jr., Factorizations of b ⁿ ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n) , 3rd ed. Providence, RI: Amer. Math. Soc., 1988.
↑ RP Brent and HJJ te Riele, Factorizations of a ⁿ ± 1, 13≤a <100 Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.

Links

Weisstein, Eric W. Cunningham Number on Wolfram MathWorld .
Sequence

A080262 in OEIS : Cunningham numbers = Cunningham numbers

The Cunningham Project, a collaborative effort to factor Cunningham numbers (neopr.) .

[mw-1] ¹ ² ³ ⁴ Weisstein, Eric W. Cunningham Number on the Wolfram MathWorld website.

[nap-2] ¹ ² Giovanni Resta. Cunningham numbers (neopr.) . Numbers Aplenty .

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

[4] RP Brent and HJJ te Riele, Factorizations of a ⁿ ± 1, 13≤a <100 Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.