Dozenal Wiki
Advertisement

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.[1] 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:[2]
  • 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)) [3]
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)
where is theth Lucas number, is theth Fibonacci number.

History[]

In 10EE, Aurifeuille discovered the factorization of

for k = 12 as the following:[2][5]

The second factor is prime, and the factorization of the first factor is

.[5] The general form of the factorization was later discovered by Lucas.[2]

  1. Template:Cite journal
  2. 2.0 2.1 2.2 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
  5. 5.0 5.1 Integer Arithmetic, Number Theory – Aurifeuillian Factorizations, Numericana