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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).
\end{displaymath} (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
$\displaystyle 2^{4k-2}+1$ $\textstyle =$ $\displaystyle (2^{2k-1}-2^k+1)(2^{2k-1}+2^k+1)$ (2)
$\displaystyle 3^{6k-3}+1$ $\textstyle =$ $\displaystyle (3^{2k-1}+1)(3^{2k-1}-3^k+1)(3^{2k-1}+3^k+1),$  

which can be written
$\displaystyle 2^{2h}+1$ $\textstyle =$ $\displaystyle L_{2h}M_{2h}$ (4)
$\displaystyle 3^{3h}+1$ $\textstyle =$ $\displaystyle (3^h+1)L_{3h}M_{3h}$ (5)
$\displaystyle 5^{5h}-1$ $\textstyle =$ $\displaystyle (5^h-1)L_{5h}M_{5h},$ (6)

where $h\equiv 2k-1$ and
$\displaystyle L_{2h}, M_{2h}$ $\textstyle =$ $\displaystyle 2^h+1\mp 2^k$ (7)
$\displaystyle L_{3h}, M_{3h}$ $\textstyle =$ $\displaystyle 3^h+1\mp 3^k$ (8)
$\displaystyle L_{5h}, M_{5h}$ $\textstyle =$ $\displaystyle 5^{2h}+3\cdot 5^h+1\mp 5^k(5^h+1).$ (9)

See also Gauss's Formula


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.

© 1996-9 Eric W. Weisstein