In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless, certain families of integers coming from cyclotomic polynomials have factorizations given by formulas applying to the whole family, as in the examples below.

Examples

!b !Number !(C − D) * (C + D) = L * M !F !C !D !2 !3 !5 !6 !7 !10 + 5(102k + 1) + 1

• 10k + 1 !11 - 114k + 2 + 5(112k + 1) + 1
• 113k + 2 + 11k + 1 !12 !13 + 19(136k + 3) + 15(134k + 2) + 7(132k + 1) + 1
• 5(135k + 3) + 3(133k + 2) + 13k + 1 !14 - 7(146k + 3) + 3(144k + 2) + 7(142k + 1) + 1

• 145k + 3 + 2(143k + 2) + 14k + 1 !15 + 154k + 2 - 152k + 1 + 1

• 8(152k + 1) + 1
• 15k + 1 !17 - 5(1710k + 5) - 15(178k + 4) - 5(176k + 3)
• 11(174k + 2) + 9(172k + 1) + 1

• 3(179k + 5) - 3(177k + 4) + 175k + 3

• 3(173k + 2) + 17k + 1 !18 !19 + 27(1912k + 6) + 31(1910k + 5) + 31(198k + 4)
• 27(196k + 3) + 17(194k + 2) + 9(192k + 1) + 1
• 7(1911k + 6) + 7(199k + 5) + 7(197k + 4)
• 5(195k + 3) + 3(193k + 2) + 19k + 1 !20 !21 - 214k + 2 - 212k + 1 - 1
• 7(216k + 3) + 13(214k + 2) + 10(212k + 1) + 1
• 2(215k + 3) + 3(213k + 2) + 21k + 1 !22 + 33(2214k + 7) + 21(2212k + 6) + 11(2210k + 5)
• 21(228k + 4) + 33(226k + 3) + 27(224k + 2)
• 11(222k + 1) + 1
• 6(2213k + 7) + 3(2211k + 6) + 3(229k + 5)
• 6(227k + 4) + 7(225k + 3) + 4(223k + 2)
• 22k + 1 !23 - 19(2316k + 8) - 15(2314k + 7) + 25(2312k + 6)
• 25(2310k + 5) - 15(238k + 4) - 19(236k + 3)
• 9(234k + 2) + 11(232k + 1) + 1

• 5(2315k + 8) + 2313k + 7 + 7(2311k + 6)

• 239k + 5 - 5(237k + 4) - 235k + 3
• 3(233k + 2) + 23k + 1 !24

History

In 1869, before the discovery of aurifeuillean factorizations, Fortuné Landry, through a tremendous manual effort, obtained the following factorization into primes: Three years later, in 1871, Aurifeuille discovered the nature of this factorization; the number 2^{4k+2}+1 for k=14, with the formula from the previous section, factors as: Of course, Landry's full factorization follows from this (taking out the obvious factor of 5). The general form of the factorization was later discovered by Lucas. 536903681 is an example of a Gaussian Mersenne norm.