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Cunningham Project

The current limits of the exponents are:

Factors of Cunningham number[edit]

Two types of factors can be derived from a Cunningham number without having to use a factorization algorithm: algebraic factorsofbinomial numbers (e.g. difference of two squares and sum of two cubes), which depend on the exponent, and aurifeuillean factors, which depend on both the base and the exponent.

Algebraic factors[edit]

From elementary algebra,

${\displaystyle (b^{kn}-1)=(b^{n}-1)\sum _{r=0}^{k-1}b^{rn}}$

for all k, and

${\displaystyle (b^{kn}+1)=(b^{n}+1)\sum _{r=0}^{k-1}(-1)^{r}\cdot b^{rn}}$

for odd k. In addition, b²ⁿ − 1 = (bⁿ − 1)(bⁿ + 1). Thus, when m divides n, b^m − 1 and b^m + 1 are factors of bⁿ − 1 if the quotient of n over miseven; only the first number is a factor if the quotient is odd. b^m + 1 is a factor of bⁿ − 1, if m divides n and the quotient is odd.

In fact,

${\displaystyle b^{n}-1=\prod _{d\mid n}\Phi _{d}($b)}

and

${\displaystyle b^{n}+1=\prod _{d\mid 2n,\,d\nmid n}\Phi _{d}($b)}

See this page for more information.

Aurifeuillean factors[edit]

When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the Cunningham project bases as a product of F, L and M:^[4]

Let b = s² × k with squarefree k, if one of the conditions holds, then ${\displaystyle \Phi _{n}($b)} have aurifeuillean factorization.

(i) ${\displaystyle k\equiv 1\mod 4}$ and ${\displaystyle n\equiv k{\pmod {2k}};}$

(ii) ${\displaystyle k\equiv 2,3{\pmod {4}}}$ and ${\displaystyle n\equiv 2k{\pmod {4k}}.}$

Other factors[edit]

Once the algebraic and aurifeuillean factors are removed, the other factors of bⁿ ± 1 are always of the form 2kn + 1, since the factors of bⁿ − 1 are all factors of ${\displaystyle \Phi _{n}($b)} , and the factors of bⁿ + 1 are all factors of ${\displaystyle \Phi _{2n}($b)} . When nisprime, both algebraic and aurifeuillean factors are not possible, except the trivial factors (b − 1 for bⁿ − 1 and b + 1 for bⁿ + 1). For Mersenne numbers, the trivial factors are not possible for prime n, so all factors are of the form 2kn + 1. In general, all factors of (bⁿ − 1) /(b − 1) are of the form 2kn + 1, where b ≥ 2 and n is prime, except when n divides b − 1, in which case (bⁿ − 1)/(b − 1) is divisible by n itself.

Cunningham numbers of the form bⁿ − 1 can only be prime if b = 2 and n is prime, assuming that n ≥ 2; these are the Mersenne numbers. Numbers of the form bⁿ + 1 can only be prime if b is even and n is a power of 2, again assuming n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2. Any factor of a Fermat number 2²ⁿ + 1 is of the form k2ⁿ⁺² + 1.

Notation[edit]

bⁿ − 1 is denoted as b,n−. Similarly, bⁿ + 1 is denoted as b,n+. When dealing with numbers of the form required for aurifeuillean factorization, b,nL and b,nM are used to denote L and M in the products above.^[5] References to b,n− and b,n+ are to the number with all algebraic and aurifeuillean factors removed. For example, Mersenne numbers are of the form 2,n− and Fermat numbers are of the form 2,2ⁿ+; the number Aurifeuille factored in 1871 was the product of 2,58L and 2,58M.

See also[edit]

• Cunningham number
• ECMNET and NFS@Home, two collaborations working for the Cunningham project

References[edit]

External links[edit]

• Cunningham project homepage
• Factorizations of bⁿ±1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, second edition
• Factorizations of bⁿ±1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, third edition
• Machine-readable Cunningham tables
• The Cunningham Project
• Brent-Montgomery-te Riele table (Cunningham tables for higher bases (bases 13 ≤ b ≤ 99, perfect powers excluded, since a power of bⁿ is also a power of b))
• Online factor collection
• Cunningham project on Prime Wiki
• Cunningham projectonPrimePages