In number theory, an aurifeuillean factorization, or aurifeuillian factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.

## Examples

• Numbers of the form have the following aurifeuillean factorization: • Setting and , one obtains the following aurifeuillean factorization of : • The repunit number R10k+6 (i.e. the repunits with length end with 6) has the following aurifeuillean factorization: n left aurifeuillean primitive part of Rn right aurifeuillean primitive part of Rn 6 7 17 16 E61 1061 26 27 × 2E1 18787 36 76E077 7 × 37 × 9X17 46 91 × 13X29831 1X7 × 347 × 1X761 56 57 × 147 × X11 × 3207 76E4545077 66 5E7 × 3547080331 221 × 35X1236E57 76 131 × 91E0577131 1062061062061 86 217 × 367XE43X31E761 87 × 24915461720E701 96 17 × 49535104455502961 171 × 30351 × 43404X297987 X6 X7 × 22671E17 × 5X9815491 1062060EEE001062061 E6 E7 × 1E1 × 4X57 × 7671581 × 374594E7 76E45076E4545076E45077 106 107 × 1377 × 699931 × 134X9E4141 170897 × 76E7X58115731E7 116 117 × 577 × E31 × 200E47770064604E4391 1000000000000060000000000001 126 76E45076E4507676E45076E45077 488X011E7 × 4426577365353XE77731 136 3021 × 483E6XX387 × 6516131X106515831 186X35186X35186E186X35186X35187 146 45223370E30X917 × 29X7397103340117 1877 × 68438083EXE138E929045614677 156 6E827399X55555499X182607 X27 × 26E481 × 882X6337E9955891 166 167 × 1351 × 5952281 × 19527401825958304E55X2611 9777 × 953145X3X2XX21X90691164502498E01 176 1062060EEXE59E5E000EEXE59E5E001062061 661 × 8E4068X91 × 2459E43848X107X5E6216E331 186 5331326X4640131 × 1530425E224777317691X009X7 615275X07 × 362X336384077 × E563824503500E59E97 196 27257 × 10E7E67 × X311E323X1 × 31X1E917482870X5931367 2X07 × 5E81 × 12E21 × 111627 × 28EE971 × 3X85149473962867E737 1X6 391 × 13131691 × 265482442444878940235X69461 34391 × 26E61047 × 5821X07347 × 2E1137435735E31 1E6 186X35186X35186X35186X3435186X35186X35186X35187 1E7 × 2547 × 79811 × 10533987 × 41824821 × 6929695496913E3159811

An open problem is whether there exists n > 16 such that the two aurifeuillean primitive parts of (n = 6 mod 10, i.e. end with 6) are both primes. (note that for the two aurifeuillean primitive parts of ( in base 2) (n = 4 mod 8) are both primes for n = X, 12, 16, 1X, 26, 36, 46, 4X, 56, 5X, 92, X6, E6, 326, 3E2, ... (and both units for n = 2, and "one is prime, another is unit" for n = 6))

• Numbers of the form or , where with square-free , have aurifeuillean factorization if and only if one of the following conditions holds:
• and • and Thus, when with square-free , and is congruent to modulo , then if is congruent to 1 mod 4 (i.e. end with 1, 5, or 9), have aurifeuillean factorization, otherwise, 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 (the Cunningham project only has 2 ≤ bases ≤ 10 (perfect powers excluded, since a power of bn is also a power of b)) 
If we let L = CD, M = C + D, the Aurifeuillian factorizations for bn ± 1 of the form F * (CD) * (C + D) = F * L * M with bases 2 ≤ b ≤ 20 (perfect powers excluded, since a power of bn is also a power of b) are:
b Number (CD) * (C + D) = L * M F C D
2 24k + 2 + 1 1 22k + 1 + 1 2k + 1
3 36k + 3 + 1 32k + 1 + 1 32k + 1 + 1 3k + 1
5 5Xk + 5 - 1 52k + 1 - 1 54k + 2 + 3(52k + 1) + 1 53k + 2 + 5k + 1
6 610k + 6 + 1 64k + 2 + 1 64k + 2 + 3(62k + 1) + 1 63k + 2 + 6k + 1
7 712k + 7 + 1 72k + 1 + 1 76k + 3 + 3(74k + 2) + 3(72k + 1) + 1 75k + 3 + 73k + 2 + 7k + 1
X X18k + X + 1 X4k + 2 + 1 X8k + 4 + 5(X6k + 3) + 7(X4k + 2)
+ 5(X2k + 1) + 1
X7k + 4 + 2(X5k + 3) + 2(X3k + 2)
+ Xk + 1
E E1Xk + E + 1 E2k + 1 + 1 EXk + 5 + 5(E8k + 4) - E6k + 3
- E4k + 2 + 5(E2k + 1) + 1
E9k + 5 + E7k + 4 - E5k + 3
+ E3k + 2 + Ek + 1
10 106k + 3 + 1 102k + 1 + 1 102k + 1 + 1 6(10k)
11 1122k + 11 - 1 112k + 1 - 1 1110k + 6 + 7(11Xk + 5) + 13(118k + 4)
+ 17(116k + 3) + 13(114k + 2) + 7(112k + 1) + 1
11Ek + 6 + 3(119k + 5) + 5(117k + 4)
+ 5(115k + 3) + 3(113k + 2) + 11k + 1
12 1224k + 12 + 1 124k + 2 + 1 1210k + 6 + 7(12Xk + 5) + 3(128k + 4)
- 7(126k + 3) + 3(124k + 2) + 7(122k + 1) + 1
12Ek + 6 + 2(129k + 5) - 127k + 4
- 125k + 3 + 2(123k + 2) + 12k + 1
13 1326k + 13 + 1 1312k + 7 - 1310k + 6 + 13Xk + 5
+ 134k + 2 - 132k + 1 + 1
138k + 4 + 8(136k + 3) + 11(134k + 2)
+ 8(132k + 1) + 1
137k + 4 + 3(135k + 3) + 3(133k + 2)
+ 13k + 1
15 152Xk + 15 - 1 152k + 1 - 1 1514k + 8 + 9(1512k + 7) + E(1510k + 6)
- 5(15Xk + 5) - 13(158k + 4) - 5(156k + 3)
+ E(154k + 2) + 9(152k + 1) + 1
1513k + 8 + 3(1511k + 7) + 15Ek + 6
- 3(159k + 5) - 3(157k + 4) + 155k + 3
+ 3(153k + 2) + 15k + 1
16 164k + 2 + 1 1 162k + 1 + 1 6(16k)
17 1732k + 17 + 1 172k + 1 + 1 1716k + 9 + 9(1714k + 8) + 15(1712k + 7)
+ 23(1710k + 6) + 27(17Xk + 5) + 27(178k + 4)
+ 23(176k + 3) + 15(174k + 2) + 9(172k + 1) + 1
1715k + 9 + 3(1713k + 8) + 5(1711k + 7)
+ 7(17Ek + 6) + 7(179k + 5) + 7(177k + 4)
+ 5(175k + 3) + 3(173k + 2) + 17k + 1
18 18Xk + 5 - 1 182k + 1 - 1 184k + 2 + 3(182k + 1) + 1 X(183k + 1) + X(18k)
19 1936k + 19 - 1 1916k + 9 + 1914k + 8 + 1912k + 7
- 194k + 2 - 192k + 1 - 1
1910k + 6 + X(19Xk + 5) + 11(198k + 4)
+ 7(196k + 3) + 11(194k + 2) + X(192k + 1) + 1
19Ek + 6 + 3(199k + 5) + 2(197k + 4)
+ 2(195k + 3) + 3(193k + 2) + 19k + 1
1X 1X38k + 1X + 1 1X4k + 2 + 1 1X18k + X + E(1X16k + 9) + 23(1X14k + 8)
+ 29(1X12k + 7) + 19(1X10k + 6) + E(1XXk + 5)
+ 19(1X8k + 4) + 29(1X6k + 3) + 23(1X4k + 2)
+ E(1X2k + 1) + 1
1X17k + X + 4(1X15k + 9) + 7(1X13k + 8)
+ 6(1X11k + 7) + 3(1XEk + 6) + 3(1X9k + 5)
+ 6(1X7k + 4) + 7(1X5k + 3) + 4(1X3k + 2)
+ 1Xk + 1
1E 1E3Xk + 1E + 1 1E2k + 1 + 1 1E1Xk + E + E(1E18k + X) + 9(1E16k + 9)
- 17(1E14k + 8) - 13(1E12k + 7) + 21(1E10k + 6)
+ 21(1EXk + 5) - 13(1E8k + 4) - 17(1E6k + 3)
+ 9(1E4k + 2) + E(1E2k + 1) + 1
1E19k + E + 3(1E17k + X) - 1E15k + 9
- 5(1E13k + 8) + 1E11k + 7 + 7(1EEk + 6)
+ 1E9k + 5 - 5(1E7k + 4) - 1E5k + 3
+ 3(1E3k + 2) + 1Ek + 1
20 2010k + 6 + 1 204k + 2 + 1 204k + 2 + 3(202k + 1) + 1 10(203k + 1) + 10(20k)
• Lucas numbers have the following aurifeuillean factorization: where is the th Lucas number, is the th Fibonacci number.

## History

In 10EE, Aurifeuille discovered the factorization of for k = 12 as the following: The second factor is prime, and the factorization of the first factor is . The general form of the factorization was later discovered by Lucas.

1. Template:Cite journal
2. Template:MathWorld
3. Template:Cite web At the end of tables 2LM, 3+, 5-, 6+, 7+, X+, E+ and 10+ are formulae detailing the Aurifeuillian factorisations.
4. Lucas Aurifeuilliean primitive part