Cunningham project

The Cunningham project is a project, started in 1925, to factor numbers of the form bⁿ ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall.^[1] There are three printed versions of the table, the most recent published in 2002,^[2] as well as an online version.^[3]

The current limits of the exponents are:

Base	2	3	5	6	7	10	11	12
Limit	1300	850	550	500	450	400	350	350
Aurifeuillian limit	2600	1700	1100	1000	900	800	700	700

Factors of Cunningham numbers

Two types of factors can be derived from a Cunningham number without having to use a factorisation algorithm: algebraic factors, which depend on the exponent, and Aurifeuillian factors, which depend on both the base and the exponent.

Algebraic factors

From elementary algebra,

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

for all k, and

(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 m is even; 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,

b^{n}-1=\prod _{{d|n}}\Phi _{d}(b)

and

b^{n}+1=\prod _{{d|2n,d!|n}}\Phi _{d}(b)

Aurifeuillian factors

Main article: Aurifeuillian factorization

When the number is of a particular form (the exact expression varies with the base), Aurifeuillian factorization may be used, which gives a product of two or three numbers. The following equations give Aurifeuillian 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 $\Phi _{n}(b)$ have Aurifeuillian factorization.

(i)

k\equiv 1\mod 4

and

n\equiv k{\pmod {2k}};

(ii)

k\equiv 2,3{\pmod 4}

and

n\equiv 2k{\pmod {4k}}.

b	Number	F	L	M	Other definitions
2	2^{4k + 2} + 1	1	2^{2k + 1} - 2^{k + 1} + 1	2^{2k + 1} + 2^{k + 1} + 1
3	3^{6k + 3} + 1	3^{2k + 1} + 1	3^{2k + 1} - 3^{k + 1} + 1	3^{2k + 1} + 3^{k + 1} + 1
5	5^{10k + 5} - 1	5^{2k + 1} - 1	T² - 5^{k + 1}T + 5^{2k + 1}	T² + 5^{k + 1}T + 5^{2k + 1}	T = 5^{2k + 1} + 1
6	6^{12k + 6} + 1	6^{4k + 2} + 1	T² - 6^{'k + 1}T + 6²k + 1	T² + 6^{k + 1}T + 6^{2k + 1}	T = 6^{2k + 1} + 1
7	7^{14k + 7} + 1	7^{2k + 1} + 1	A - B	A + B	A = 7^{6k + 3} + 3(7^{4k + 2}) + 3(7^{2k + 1}) + 1 B = 7^{5k + 3} + 7^{3k + 2} + 7^{k + 1}
10	10^{20k + 10} + 1	10^{4k + 2} + 1	A - B	A + B	A = 10^{8k + 4} + 5(10^{6k + 3}) + 7(10^{4k + 2}) + 5(10^{2k + 1}) + 1 B = 10^{7k + 4} + 2(10^{5k + 3}) + 2(10^{3k + 2}) + 10^{k + 1}
11	11^{22k + 11} + 1	11^{2k + 1} + 1	A - B	A + B	A = 11^{10k + 5} + 5(11^{8k + 4}) - 11^{6k + 3} - 11^{4k + 2} + 5(11^{2k + 1}) + 1 B = 11^{9k + 5} + 11^{7k + 4} - 11^{5k + 3} + 11^{3k + 2} + 11^{k + 1}
12	12^{6k + 3} + 1	12^{2k + 1} + 1	12^{2k + 1} - 6(12^k) + 1	12^{2k + 1} + 6(12^k) + 1

Other factors

Once the algebraic and Aurifeuillian factors are removed, the other factors of bⁿ ± 1 are always of the form 2kn + 1, since they are all factors of $\Phi _{n}(b)$ . When n is prime, both algebraic and Aurifeuillian 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 a = 1. Any factor of a Fermat number 2^2ⁿ + 1 is of the form k2^n + 2 + 1.

Notation

bⁿ − 1 is denoted as b,n−. Similarly, bⁿ + 1 is denoted as b,n+. When dealing with numbers of the form required for Aurifeuillian factorisation, 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 Aurifeuillian 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.

References

↑ Cunningham, Allan J. C.; Woodall, H. J. (1925). Factorisation of yⁿ ± 1, y = 2, 3, 5, 6, 7, 10, 11, 12, up to high powers n. Hodgson.
↑ Brillhart, John; Lehmer, Derrick H.; Selfridge, John L.; Tuckerman, Bryant; Wagstaff, Samuel S. (2002). "Factorizations of bⁿ ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers". AMS.
↑ "The Cunningham Project". Retrieved 18 March 2012.
↑ "Main Cunningham Tables". Retrieved 18 March 2012. At the end of tables 2LM, 3+, 5-, 7+, 10+, 11+ and 12+ are formulae detailing the Aurifeuillian factorisations.
↑ "Explanation of the notation on the Pages". Retrieved 18 March 2012.

External links

Cunningham project homepage
Brent-Montgomery-te Riele table (Cunningham tables for higher bases)
Cunningham tables on Mersennewiki

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