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# Aurifeuillian factor

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An Aurifeuillian factor was named after the French mathematician Léon-François-Antoine Aurifeuille.

• Numbers of the form $\displaystyle{ 2^{4k+2}+1 }$ have the following Aurifeuillian factorization: Mathworld
$\displaystyle{ 2^{4k+2}+1 = (2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1) }$
• Numbers of the form $\displaystyle{ b^n - 1 }$ or $\displaystyle{ \Phi_n(b) }$, where $\displaystyle{ b = s^2 \cdot t }$ with square-free $\displaystyle{ t }$, have aurifeuillean factorization if and only if one of the following conditions holds:
• $\displaystyle{ t\equiv 1 \pmod 4 }$ and $\displaystyle{ n\equiv t \pmod{2t} }$
• $\displaystyle{ t\equiv 2, 3 \pmod 4 }$ and $\displaystyle{ n\equiv 2t \pmod{4t} }$
Thus, when $\displaystyle{ b = s^2\cdot t }$ with square-free $\displaystyle{ t }$, and $\displaystyle{ n }$ is congruent to $\displaystyle{ t }$ mod $\displaystyle{ 2t }$, then if $\displaystyle{ t }$ is congruent to 1 mod 4, $\displaystyle{ b^n-1 }$ have aurifeuillean factorization, otherwise, $\displaystyle{ b^n+1 }$ have aurifeuillean 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: Main Cunningham Tables. At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ are formulae detailing the Aurifeuillian factorisations.</ref>

If we let L = C - D, M = C + D, the Aurifeuillian factorizations for bn ± 1 with the bases 2 ≥ b ≥ 24 (perfect powers excluded, since a power of bn is also a power of b) are: (for the coefficients of the polynomials for all square-free bases up to 199, see Coefficients of Lucas C,D polynomials for all square-free bases up to 199)

(Number = F * (C - D) * (C + D) = F * L * M)

b Number (C - D) * (C + D) = L * M F C D
2 24k + 2 + 1 $\displaystyle{ \Phi_4(2^{2k+1}) }$ 1 22k + 1 + 1 2k + 1
3 36k + 3 + 1 $\displaystyle{ \Phi_6(3^{2k+1}) }$ 32k + 1 + 1 32k + 1 + 1 3k + 1
5 510k + 5 - 1 $\displaystyle{ \Phi_5(5^{2k+1}) }$ 52k + 1 - 1 54k + 2 + 3(52k + 1) + 1 53k + 2 + 5k + 1
6 612k + 6 + 1 $\displaystyle{ \Phi_{12}(6^{2k+1}) }$ 64k + 2 + 1 64k + 2 + 3(62k + 1) + 1 63k + 2 + 6k + 1
7 714k + 7 + 1 $\displaystyle{ \Phi_{14}(7^{2k+1}) }$ 72k + 1 + 1 76k + 3 + 3(74k + 2) + 3(72k + 1) + 1 75k + 3 + 73k + 2 + 7k + 1
10 1020k + 10 + 1 $\displaystyle{ \Phi_{20}(10^{2k+1}) }$ 104k + 2 + 1 108k + 4 + 5(106k + 3) + 7(104k + 2)
+ 5(102k + 1) + 1
107k + 4 + 2(105k + 3) + 2(103k + 2)
+ 10k + 1
11 1122k + 11 + 1 $\displaystyle{ \Phi_{22}(11^{2k+1}) }$ 112k + 1 + 1 1110k + 5 + 5(118k + 4) - 116k + 3
- 114k + 2 + 5(112k + 1) + 1
119k + 5 + 117k + 4 - 115k + 3
+ 113k + 2 + 11k + 1
12 126k + 3 + 1 $\displaystyle{ \Phi_6(12^{2k+1}) }$ 122k + 1 + 1 122k + 1 + 1 6(12k)
13 1326k + 13 - 1 $\displaystyle{ \Phi_{13}(13^{2k+1}) }$ 132k + 1 - 1 1312k + 6 + 7(1310k + 5) + 15(138k + 4)
+ 19(136k + 3) + 15(134k + 2) + 7(132k + 1) + 1
1311k + 6 + 3(139k + 5) + 5(137k + 4)
+ 5(135k + 3) + 3(133k + 2) + 13k + 1
14 1428k + 14 + 1 $\displaystyle{ \Phi_{28}(14^{2k+1}) }$ 144k + 2 + 1 1412k + 6 + 7(1410k + 5) + 3(148k + 4)
- 7(146k + 3) + 3(144k + 2) + 7(142k + 1) + 1
1411k + 6 + 2(149k + 5) - 147k + 4
- 145k + 3 + 2(143k + 2) + 14k + 1
15 1530k + 15 + 1 $\displaystyle{ \Phi_{30}(15^{2k+1}) }$ 1514k + 7 - 1512k + 6 + 1510k + 5
+ 154k + 2 - 152k + 1 + 1
158k + 4 + 8(156k + 3) + 13(154k + 2)
+ 8(152k + 1) + 1
157k + 4 + 3(155k + 3) + 3(153k + 2)
+ 15k + 1
17 1734k + 17 - 1 $\displaystyle{ \Phi_{17}(17^{2k+1}) }$ 172k + 1 - 1 1716k + 8 + 9(1714k + 7) + 11(1712k + 6)
- 5(1710k + 5) - 15(178k + 4) - 5(176k + 3)
+ 11(174k + 2) + 9(172k + 1) + 1
1715k + 8 + 3(1713k + 7) + 1711k + 6
- 3(179k + 5) - 3(177k + 4) + 175k + 3
+ 3(173k + 2) + 17k + 1
18 184k + 2 + 1 $\displaystyle{ \Phi_4(18^{2k+1}) }$ 1 182k + 1 + 1 6(18k)
19 1938k + 19 + 1 $\displaystyle{ \Phi_{38}(19^{2k+1}) }$ 192k + 1 + 1 1917k + 9 + 3(1915k + 8) + 5(1913k + 7)
+ 7(1911k + 6) + 7(199k + 5) + 7(197k + 4)
+ 5(195k + 3) + 3(193k + 2) + 19k + 1
20 2010k + 5 - 1 $\displaystyle{ \Phi_5(20^{2k+1}) }$ 202k + 1 - 1 204k + 2 + 3(202k + 1) + 1 10(203k + 1) + 10(20k)
21 2142k + 21 - 1 $\displaystyle{ \Phi_{21}(21^{2k+1}) }$ 2118k + 9 + 2116k + 8 + 2114k + 7
- 214k + 2 - 212k + 1 - 1
2112k + 6 + 10(2110k + 5) + 13(218k + 4)
+ 7(216k + 3) + 13(214k + 2) + 10(212k + 1) + 1
2111k + 6 + 3(219k + 5) + 2(217k + 4)
+ 2(215k + 3) + 3(213k + 2) + 21k + 1
22 2244k + 22 + 1 $\displaystyle{ \Phi_{44}(22^{2k+1}) }$ 224k + 2 + 1 2220k + 10 + 11(2218k + 9) + 27(2216k + 8)
+ 33(2214k + 7) + 21(2212k + 6) + 11(2210k + 5)
+ 21(228k + 4) + 33(226k + 3) + 27(224k + 2)
+ 11(222k + 1) + 1
2219k + 10 + 4(2217k + 9) + 7(2215k + 8)
+ 6(2213k + 7) + 3(2211k + 6) + 3(229k + 5)
+ 6(227k + 4) + 7(225k + 3) + 4(223k + 2)
+ 22k + 1
23 2346k + 23 + 1 $\displaystyle{ \Phi_{46}(23^{2k+1}) }$ 232k + 1 + 1 2322k + 11 + 11(2320k + 10) + 9(2318k + 9)
- 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
2321k + 11 + 3(2319k + 10) - 2317k + 9
- 5(2315k + 8) + 2313k + 7 + 7(2311k + 6)
+ 239k + 5 - 5(237k + 4) - 235k + 3
+ 3(233k + 2) + 23k + 1
24 2412k + 6 + 1 $\displaystyle{ \Phi_{12}(24^{2k+1}) }$ 244k + 2 + 1 244k + 2 + 3(242k + 1) + 1 12(243k + 1) + 12(24k)
(See List of Aurifeuillean factorization for more information (square-free bases up to 199))
• Numbers of the form $\displaystyle{ a^4 + 4b^4 }$ have the following aurifeuillean factorization:
$\displaystyle{ a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2) }$
$\displaystyle{ L_{10k+5} = L_{2k+1}\cdot (5{F_{2k+1}}^2-5F_{2k+1}+1)\cdot (5{F_{2k+1}}^2+5F_{2k+1}+1) }$
where $\displaystyle{ L_n }$ is the $\displaystyle{ n }$th Lucas number, $\displaystyle{ F_n }$ is the $\displaystyle{ n }$th Fibonacci number.