Aurifeuillean Factorization

A factorization of the form

$$2^{4n+2}+1 = (2^{2n+1}-2^{n+1}+1)(2^{2n+1}+2^{n+1}+1).$$ (1)

The factorization for $n=14$ was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as $L$ and $M$ as follows

$$2^{4k-2}+1 = (2^{2k-1}-2^k+1)(2^{2k-1}+2^k+1)$$ (2)

$$3^{6k-3}+1 = (3^{2k-1}+1)(3^{2k-1}-3^k+1)(3^{2k-1}+3^k+1),$$ (3)

which can be written

$$2^{2h}+1 = L_{2h}M_{2h}$$ (4)

$$3^{3h}+1 = (3^h+1)L_{3h}M_{3h}$$ (5)

$$5^{5h}-1 = (5^h-1)L_{5h}M_{5h},$$ (6)

where $h\equiv 2k-1$ and

$$L_{2h}, M_{2h} = 2^h+1\mp 2^k$$ (7)

$$L_{3h}, M_{3h} = 3^h+1\mp 3^k$$ (8)

$$L_{5h}, M_{5h} = 5^{2h}+3\cdot 5^h+1\mp 5^k(5^h+1).$$ (9)

See also Gauss's Formula

References

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

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