Aurifeuillean Factorization

A factorization of the form

(1)

The factorization for  was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as  and  as follows:

			(2)
			(3)

which can be written

			(4)
			(5)
			(6)

where  and

			(7)
			(8)
			(9)

REFERENCES:

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., pp. lxviii-lxxii, 1988.

Riesel, H. "Aurifeullian Factorization" in Appendix 6. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 309-315, 1994.

Wagstaff, S. S. Jr. "Aurifeullian Factorizations and the Period of the Bell Numbers Modulo a Prime." Math. Comput. 65, 383-391, 1996.