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Difference between revisions of "Aurifeuillian factor"
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| − | *Numbers of the form <math>2^{4k+2}+1</math> have the following ''' | + | {{Spelling|Aurifeuillean }} |
| + | An '''Aurifeuillian factor''' was named after the French mathematician [[Léon-François-Antoine Aurifeuille]]. | ||
| + | |||
| + | *Numbers of the form <math>2^{4k+2}+1</math> have the following '''Aurifeuillian factorization''': [http://mathworld.wolfram.com/AurifeuilleanFactorization.html Mathworld] | ||
::<math>2^{4k+2}+1 = (2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1)</math> | ::<math>2^{4k+2}+1 = (2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1)</math> | ||
*Numbers of the form <math>b^n - 1</math> or <math>\Phi_n(b)</math>, where <math>b = s^2 \cdot t</math> with [[Square-free integer|square-free]] <math>t</math>, have aurifeuillean factorization if and only if one of the following conditions holds: | *Numbers of the form <math>b^n - 1</math> or <math>\Phi_n(b)</math>, where <math>b = s^2 \cdot t</math> with [[Square-free integer|square-free]] <math>t</math>, have aurifeuillean factorization if and only if one of the following conditions holds: | ||
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:Thus, when <math>b = s^2\cdot t</math> with square-free <math>t</math>, and <math>n</math> is [[Congruence relation|congruent]] to <math>t</math> mod <math>2t</math>, then if <math>t</math> is congruent to 1 mod 4, <math>b^n-1</math> have aurifeuillean factorization, otherwise, <math>b^n+1</math> have aurifeuillean factorization. | :Thus, when <math>b = s^2\cdot t</math> with square-free <math>t</math>, and <math>n</math> is [[Congruence relation|congruent]] to <math>t</math> mod <math>2t</math>, then if <math>t</math> is congruent to 1 mod 4, <math>b^n-1</math> have aurifeuillean factorization, otherwise, <math>b^n+1</math> 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'': [http://homes.cerias.purdue.edu/~ssw/cun/pmain1215 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 ''b''<sup>''n''</sup> ± 1 with the bases 2 ≥ ''b'' ≥ 24 ([[perfect power]]s excluded, since a power of ''b''<sup>''n''</sup> is also a power of ''b'') are: (for the coefficients of the polynomials for all square-free bases up to 199, see [http://myfactorcollection.mooo.com:8090/LCD_2_199 Coefficients of Lucas C,D polynomials for all square-free bases up to 199]) | |
| − | + | (Number = ''F'' * (''C'' - ''D'') * (''C'' + ''D'') = ''F'' * ''L'' * ''M'') | |
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| − | :(See [ | + | :(See [https://stdkmd.com/nrr/repunit/repunitnote.htm#aurifeuillean List of Aurifeuillean factorization] for more information (square-free bases up to 199)) |
*Numbers of the form <math>a^4 + 4b^4</math> have the following aurifeuillean factorization: | *Numbers of the form <math>a^4 + 4b^4</math> have the following aurifeuillean factorization: | ||
:: <math>a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2)</math> | :: <math>a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2)</math> | ||
| − | *[[Lucas number]]s <math>L_{10k+5}</math> have the following aurifeuillean factorization: [https:// | + | *[[Lucas number]]s <math>L_{10k+5}</math> have the following aurifeuillean factorization: [https://primes.utm.edu/top20/page.php?id=21 Lucas Aurifeuilliean primitive part] |
::<math>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)</math> | ::<math>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)</math> | ||
:where <math>L_n</math> is the <math>n</math>th Lucas number, <math>F_n</math> is the <math>n</math>th [[Fibonacci number]]. | :where <math>L_n</math> is the <math>n</math>th Lucas number, <math>F_n</math> is the <math>n</math>th [[Fibonacci number]]. | ||
Latest revision as of 08:04, 24 June 2019
An Aurifeuillian factor was named after the French mathematician Léon-François-Antoine Aurifeuille.
- Numbers of the form [math]\displaystyle{ 2^{4k+2}+1 }[/math] have the following Aurifeuillian factorization: Mathworld
- [math]\displaystyle{ 2^{4k+2}+1 = (2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1) }[/math]
- Numbers of the form [math]\displaystyle{ b^n - 1 }[/math] or [math]\displaystyle{ \Phi_n(b) }[/math], where [math]\displaystyle{ b = s^2 \cdot t }[/math] with square-free [math]\displaystyle{ t }[/math], have aurifeuillean factorization if and only if one of the following conditions holds:
- [math]\displaystyle{ t\equiv 1 \pmod 4 }[/math] and [math]\displaystyle{ n\equiv t \pmod{2t} }[/math]
- [math]\displaystyle{ t\equiv 2, 3 \pmod 4 }[/math] and [math]\displaystyle{ n\equiv 2t \pmod{4t} }[/math]
- Thus, when [math]\displaystyle{ b = s^2\cdot t }[/math] with square-free [math]\displaystyle{ t }[/math], and [math]\displaystyle{ n }[/math] is congruent to [math]\displaystyle{ t }[/math] mod [math]\displaystyle{ 2t }[/math], then if [math]\displaystyle{ t }[/math] is congruent to 1 mod 4, [math]\displaystyle{ b^n-1 }[/math] have aurifeuillean factorization, otherwise, [math]\displaystyle{ b^n+1 }[/math] 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 | [math]\displaystyle{ \Phi_4(2^{2k+1}) }[/math] | 1 | 22k + 1 + 1 | 2k + 1 |
| 3 | 36k + 3 + 1 | [math]\displaystyle{ \Phi_6(3^{2k+1}) }[/math] | 32k + 1 + 1 | 32k + 1 + 1 | 3k + 1 |
| 5 | 510k + 5 - 1 | [math]\displaystyle{ \Phi_5(5^{2k+1}) }[/math] | 52k + 1 - 1 | 54k + 2 + 3(52k + 1) + 1 | 53k + 2 + 5k + 1 |
| 6 | 612k + 6 + 1 | [math]\displaystyle{ \Phi_{12}(6^{2k+1}) }[/math] | 64k + 2 + 1 | 64k + 2 + 3(62k + 1) + 1 | 63k + 2 + 6k + 1 |
| 7 | 714k + 7 + 1 | [math]\displaystyle{ \Phi_{14}(7^{2k+1}) }[/math] | 72k + 1 + 1 | 76k + 3 + 3(74k + 2) + 3(72k + 1) + 1 | 75k + 3 + 73k + 2 + 7k + 1 |
| 10 | 1020k + 10 + 1 | [math]\displaystyle{ \Phi_{20}(10^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{22}(11^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_6(12^{2k+1}) }[/math] | 122k + 1 + 1 | 122k + 1 + 1 | 6(12k) |
| 13 | 1326k + 13 - 1 | [math]\displaystyle{ \Phi_{13}(13^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{28}(14^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{30}(15^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{17}(17^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_4(18^{2k+1}) }[/math] | 1 | 182k + 1 + 1 | 6(18k) |
| 19 | 1938k + 19 + 1 | [math]\displaystyle{ \Phi_{38}(19^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_5(20^{2k+1}) }[/math] | 202k + 1 - 1 | 204k + 2 + 3(202k + 1) + 1 | 10(203k + 1) + 10(20k) |
| 21 | 2142k + 21 - 1 | [math]\displaystyle{ \Phi_{21}(21^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{44}(22^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{46}(23^{2k+1}) }[/math] | 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 | [math]\displaystyle{ \Phi_{12}(24^{2k+1}) }[/math] | 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 [math]\displaystyle{ a^4 + 4b^4 }[/math] have the following aurifeuillean factorization:
- [math]\displaystyle{ a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2) }[/math]
- Lucas numbers [math]\displaystyle{ L_{10k+5} }[/math] have the following aurifeuillean factorization: Lucas Aurifeuilliean primitive part
- [math]\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) }[/math]
- where [math]\displaystyle{ L_n }[/math] is the [math]\displaystyle{ n }[/math]th Lucas number, [math]\displaystyle{ F_n }[/math] is the [math]\displaystyle{ n }[/math]th Fibonacci number.
