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In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials.^[1] 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

Numbers of the form $a^{4}+4b^{4}$ have the following factorization (Sophie Germain's identity): $a^{4}+4b^{4}=(a^{2}-2ab+2b^{2})\cdot (a^{2}+2ab+2b^{2}).$ Setting $a=1$ and $b=2^{k}$ , one obtains the following aurifeuillean factorization of $\Phi _{4}(2^{2k+1})=2^{4k+2}+1$ , where $\Phi _{4}(x)=x^{2}+1$ is the fourth cyclotomic polynomial:^[2] $2^{4k+2}+1=(2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1).$
Numbers of the form $a^{6}+27b^{6}$ have the following factorization, where the first factor ( $a^{2}+3b^{2}$ ) is the algebraic factorization of sum of two cubes: $a^{6}+27b^{6}=(a^{2}+3b^{2})\cdot (a^{2}-3ab+3b^{2})\cdot (a^{2}+3ab+3b^{2}).$ Setting $a=1$ and $b=3^{k}$ , one obtains the following factorization of $3^{6k+3}+1$ :^[2] $3^{6k+3}+1=(3^{2k+1}+1)\cdot (3^{2k+1}-3^{k+1}+1)\cdot (3^{2k+1}+3^{k+1}+1).$ Here, the first of the three terms in the factorization is $\Phi _{2}(3^{2k+1})$ and the remaining two terms provide an aurifeuillean factorization of $\Phi _{6}(3^{2k+1})$ , where $\Phi _{6}(x)=x^{2}-x+1$ .
Numbers of the form $b^{n}-1$ $b^{n}-1$ or their factors $\Phi _{n}(b)$ $\Phi _{n}(b)$ , where $b=s^{2}\cdot t$ $b=s^{2}\cdot t$ with square-free $t$ $t$ , have aurifeuillean factorization if and only if one of the following conditions holds:
- $t\equiv 1{\pmod {4}}$ and $n\equiv t{\pmod {2t}}$
- $t\equiv 2,3{\pmod {4}}$ and $n\equiv 2t{\pmod {4t}}$

Thus, when

b=s^{2}\cdot t

with square-free

t

, and

n

is congruent to

t

modulo

2t

, then if

t

is congruent to 1 mod 4,

b^{n}-1

have aurifeuillean factorization, otherwise,

b^{n}+1

have aurifeuillean factorization.

When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the Cunningham project bases as a product of F, L and M:^[3]

If we let L = C − D, M = C + D, the aurifeuillean factorizations for bⁿ ± 1 of the form F * (C − D) * (C + D) = F * L * M 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 and up to 998, see ^[4]^[5]^[6])

b	Number	(C − D) * (C + D) = L * M	F	C	D
2	2^{4k + 2} + 1	$\Phi _{4}(2^{2k+1})$	1	2^{2k + 1} + 1	2^{k + 1}
3	3^{6k + 3} + 1	$\Phi _{6}(3^{2k+1})$	3^{2k + 1} + 1	3^{2k + 1} + 1	3^{k + 1}
5	5^{10k + 5} - 1	$\Phi _{5}(5^{2k+1})$	5^{2k + 1} - 1	5^{4k + 2} + 3(5^{2k + 1}) + 1	5^{3k + 2} + 5^{k + 1}
6	6^{12k + 6} + 1	$\Phi _{12}(6^{2k+1})$	6^{4k + 2} + 1	6^{4k + 2} + 3(6^{2k + 1}) + 1	6^{3k + 2} + 6^{k + 1}
7	7^{14k + 7} + 1	$\Phi _{14}(7^{2k+1})$	7^{2k + 1} + 1	7^{6k + 3} + 3(7^{4k + 2}) + 3(7^{2k + 1}) + 1	7^{5k + 3} + 7^{3k + 2} + 7^{k + 1}
10	10^{20k + 10} + 1	$\Phi _{20}(10^{2k+1})$	10^{4k + 2} + 1	10^{8k + 4} + 5(10^{6k + 3}) + 7(10^{4k + 2}) + 5(10^{2k + 1}) + 1	10^{7k + 4} + 2(10^{5k + 3}) + 2(10^{3k + 2}) + 10^{k + 1}
11	11^{22k + 11} + 1	$\Phi _{22}(11^{2k+1})$	11^{2k + 1} + 1	11^{10k + 5} + 5(11^{8k + 4}) - 11^{6k + 3} - 11^{4k + 2} + 5(11^{2k + 1}) + 1	11^{9k + 5} + 11^{7k + 4} - 11^{5k + 3} + 11^{3k + 2} + 11^{k + 1}
12	12^{6k + 3} + 1	$\Phi _{6}(12^{2k+1})$	12^{2k + 1} + 1	12^{2k + 1} + 1	6(12^k)
13	13^{26k + 13} - 1	$\Phi _{13}(13^{2k+1})$	13^{2k + 1} - 1	13^{12k + 6} + 7(13^{10k + 5}) + 15(13^{8k + 4}) + 19(13^{6k + 3}) + 15(13^{4k + 2}) + 7(13^{2k + 1}) + 1	13^{11k + 6} + 3(13^{9k + 5}) + 5(13^{7k + 4}) + 5(13^{5k + 3}) + 3(13^{3k + 2}) + 13^{k + 1}
14	14^{28k + 14} + 1	$\Phi _{28}(14^{2k+1})$	14^{4k + 2} + 1	14^{12k + 6} + 7(14^{10k + 5}) + 3(14^{8k + 4}) - 7(14^{6k + 3}) + 3(14^{4k + 2}) + 7(14^{2k + 1}) + 1	14^{11k + 6} + 2(14^{9k + 5}) - 14^{7k + 4} - 14^{5k + 3} + 2(14^{3k + 2}) + 14^{k + 1}
15	15^{30k + 15} + 1	$\Phi _{30}(15^{2k+1})$	15^{14k + 7} - 15^{12k + 6} + 15^{10k + 5} + 15^{4k + 2} - 15^{2k + 1} + 1	15^{8k + 4} + 8(15^{6k + 3}) + 13(15^{4k + 2}) + 8(15^{2k + 1}) + 1	15^{7k + 4} + 3(15^{5k + 3}) + 3(15^{3k + 2}) + 15^{k + 1}
17	17^{34k + 17} - 1	$\Phi _{17}(17^{2k+1})$	17^{2k + 1} - 1	17^{16k + 8} + 9(17^{14k + 7}) + 11(17^{12k + 6}) - 5(17^{10k + 5}) - 15(17^{8k + 4}) - 5(17^{6k + 3}) + 11(17^{4k + 2}) + 9(17^{2k + 1}) + 1	17^{15k + 8} + 3(17^{13k + 7}) + 17^{11k + 6} - 3(17^{9k + 5}) - 3(17^{7k + 4}) + 17^{5k + 3} + 3(17^{3k + 2}) + 17^{k + 1}
18	18^{4k + 2} + 1	$\Phi _{4}(18^{2k+1})$	1	18^{2k + 1} + 1	6(18^k)
19	19^{38k + 19} + 1	$\Phi _{38}(19^{2k+1})$	19^{2k + 1} + 1	19^{18k + 9} + 9(19^{16k + 8}) + 17(19^{14k + 7}) + 27(19^{12k + 6}) + 31(19^{10k + 5}) + 31(19^{8k + 4}) + 27(19^{6k + 3}) + 17(19^{4k + 2}) + 9(19^{2k + 1}) + 1	19^{17k + 9} + 3(19^{15k + 8}) + 5(19^{13k + 7}) + 7(19^{11k + 6}) + 7(19^{9k + 5}) + 7(19^{7k + 4}) + 5(19^{5k + 3}) + 3(19^{3k + 2}) + 19^{k + 1}
20	20^{10k + 5} - 1	$\Phi _{5}(20^{2k+1})$	20^{2k + 1} - 1	20^{4k + 2} + 3(20^{2k + 1}) + 1	10(20^{3k + 1}) + 10(20^k)
21	21^{42k + 21} - 1	$\Phi _{21}(21^{2k+1})$	21^{18k + 9} + 21^{16k + 8} + 21^{14k + 7} - 21^{4k + 2} - 21^{2k + 1} - 1	21^{12k + 6} + 10(21^{10k + 5}) + 13(21^{8k + 4}) + 7(21^{6k + 3}) + 13(21^{4k + 2}) + 10(21^{2k + 1}) + 1	21^{11k + 6} + 3(21^{9k + 5}) + 2(21^{7k + 4}) + 2(21^{5k + 3}) + 3(21^{3k + 2}) + 21^{k + 1}
22	22^{44k + 22} + 1	$\Phi _{44}(22^{2k+1})$	22^{4k + 2} + 1	22^{20k + 10} + 11(22^{18k + 9}) + 27(22^{16k + 8}) + 33(22^{14k + 7}) + 21(22^{12k + 6}) + 11(22^{10k + 5}) + 21(22^{8k + 4}) + 33(22^{6k + 3}) + 27(22^{4k + 2}) + 11(22^{2k + 1}) + 1	22^{19k + 10} + 4(22^{17k + 9}) + 7(22^{15k + 8}) + 6(22^{13k + 7}) + 3(22^{11k + 6}) + 3(22^{9k + 5}) + 6(22^{7k + 4}) + 7(22^{5k + 3}) + 4(22^{3k + 2}) + 22^{k + 1}
23	23^{46k + 23} + 1	$\Phi _{46}(23^{2k+1})$	23^{2k + 1} + 1	23^{22k + 11} + 11(23^{20k + 10}) + 9(23^{18k + 9}) - 19(23^{16k + 8}) - 15(23^{14k + 7}) + 25(23^{12k + 6}) + 25(23^{10k + 5}) - 15(23^{8k + 4}) - 19(23^{6k + 3}) + 9(23^{4k + 2}) + 11(23^{2k + 1}) + 1	23^{21k + 11} + 3(23^{19k + 10}) - 23^{17k + 9} - 5(23^{15k + 8}) + 23^{13k + 7} + 7(23^{11k + 6}) + 23^{9k + 5} - 5(23^{7k + 4}) - 23^{5k + 3} + 3(23^{3k + 2}) + 23^{k + 1}
24	24^{12k + 6} + 1	$\Phi _{12}(24^{2k+1})$	24^{4k + 2} + 1	24^{4k + 2} + 3(24^{2k + 1}) + 1	12(24^{3k + 1}) + 12(24^k)

Lucas numbers $L_{10k+5}$ have the following aurifeuillean factorization:^[7]

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

L_{n}

is the

n

th Lucas number, and

F_{n}

is the

n

th Fibonacci number.

History

In 1869, before the discovery of aurifeuillean factorizations, Landry [fr; es; de], through a tremendous manual effort,^[8]^[9] obtained the following factorization into primes:

2^{58}+1=5\cdot 107367629\cdot 536903681.

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:^[2]^[8]

2^{58}+1=(2^{29}-2^{15}+1)(2^{29}+2^{15}+1)=536838145\cdot 536903681.

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.^[2]

536903681 is an example of a Gaussian Mersenne norm.^[9]

References

^ A. Granville, P. Pleasants (2006). "Aurifeuillian factorization" (PDF). Math. Comp. 75 (253): 497–508. doi:10.1090/S0025-5718-05-01766-7.
^ ^a ^b ^c ^d Weisstein, Eric W. "Aurifeuillean Factorization". MathWorld.
^ "Main Cunningham Tables". At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ are formulae detailing the aurifeuillean factorizations.
^ List of aurifeuillean factorization of cyclotomic numbers (square-free bases up to 199)
^ Coefficients of Lucas C,D polynomials for all square-free bases up to 199
^ Coefficients of Lucas C,D polynomials for all square-free bases up to 998
^ Lucas Aurifeuilliean primitive part
^ ^a ^b Integer Arithmetic, Number Theory – Aurifeuillean Factorizations, Numericana
^ ^a ^b Gaussian Mersenne, the Prime Pages glossary

External links

Aurifeuillean Factorisation, Colin Barker
Aurifeuillean Factorizations, Gérard P. Michon
The Search for Aurifeuillean-Like Factorizations
Online factor collection
A Note on Aurifeuillean Factorizations
Aurifeuillean Factorisation

[1] A. Granville, P. Pleasants (2006). "Aurifeuillian factorization" (PDF). Math. Comp. 75 (253): 497–508. doi:10.1090/S0025-5718-05-01766-7.

[Wolfram-2] Weisstein, Eric W. "Aurifeuillean Factorization". MathWorld.

[3] "Main Cunningham Tables". At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ are formulae detailing the aurifeuillean factorizations.

[4] List of aurifeuillean factorization of cyclotomic numbers (square-free bases up to 199)

[5] Coefficients of Lucas C,D polynomials for all square-free bases up to 199

[6] Coefficients of Lucas C,D polynomials for all square-free bases up to 998

[7] Lucas Aurifeuilliean primitive part

[numericana-8] Integer Arithmetic, Number Theory – Aurifeuillean Factorizations, Numericana

[primepages-9] Gaussian Mersenne, the Prime Pages glossary

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

@@ Line 1: / Line 1: @@
-In [[number theory]], an '''aurifeuillean factorization''', or '''aurifeuillian factorization''', named after [[Léon-François-Antoine Aurifeuille]], is a special type of [[Algebraic expression|algebraic]] [[factorization]] that comes from non-trivial factorizations of [[cyclotomic polynomial]]s over the [[integer]]s.<ref>{{cite journal |url=http://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf |title=Aurifeuillian factorization |author=A. Granville, P. Pleasants |journal=Math. Comp. |volume=75 |year=2006 |pages=497–508 |doi=10.1090/S0025-5718-05-01766-7}}</ref> Although cyclotomic polynomials themselves are [[irreducible polynomial|irreducible]] over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.
+In [[number theory]], an '''aurifeuillean factorization''', named after [[Léon-François-Antoine Aurifeuille]], is [[factorization]] of certain integer values of the  [[cyclotomic polynomial]]s.<ref>{{cite journal |url=https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf |title={{not a typo|Aurifeuillian}} factorization |author=A. Granville, P. Pleasants |journal=Math. Comp. |volume=75 |issue=253 |year=2006 |pages=497–508 |doi=10.1090/S0025-5718-05-01766-7|doi-access=free }}</ref> Because cyclotomic polynomials are [[irreducible polynomial]]s 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 ==
+* Numbers of the form <math>a^4 + 4b^4</math> have the following factorization ([[Sophie Germain's identity]]): <math display=block>a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2).</math> Setting <math>a=1</math> and <math>b=2^k</math>, one obtains the following aurifeuillean factorization of <math>\Phi_4(2^{2k+1})=2^{4k+2}+1</math>, where <math>\Phi_4(x)=x^2+1</math> is the fourth cyclotomic polynomial:<ref name="Wolfram">{{MathWorld|title=Aurifeuillean Factorization|urlname=AurifeuilleanFactorization}}</ref> <math display=block>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>a^6 + 27b^6</math> have the following factorization, where the first factor (<math>a^2 + 3b^2</math>) is the algebraic factorization of [[sum of two cubes]]: <math display=block>a^6 + 27b^6 = (a^2 + 3b^2)\cdot (a^2 - 3ab + 3b^2)\cdot (a^2 + 3ab + 3b^2).</math> Setting <math>a=1</math> and <math>b=3^k</math>, one obtains the following factorization of <math>3^{6k+3}+1</math>:<ref name="Wolfram"></ref> <math display=block>3^{6k+3}+1 = (3^{2k+1}+1)\cdot (3^{2k+1}-3^{k+1}+1)\cdot (3^{2k+1}+3^{k+1}+1).</math> Here, the first of the three terms in the factorization is <math>\Phi_2(3^{2k+1})</math> and the remaining two terms provide an aurifeuillean factorization of <math>\Phi_6(3^{2k+1})</math>, where <math>\Phi_6(x)=x^2-x+1</math>.
-* Numbers of the form <math>2^{4k+2}+1</math> have the following aurifeuillean factorization:<ref name="Wolfram">{{MathWorld|title=Aurifeuillean Factorization|urlname=AurifeuilleanFactorization}}</ref>
+* Numbers of the form <math>b^n - 1</math> or their factors <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:
-:: <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:
 ** <math>t\equiv 1 \pmod 4</math> and <math>n\equiv t \pmod{2t}</math>
 ** <math>t\equiv 2, 3 \pmod 4</math> and <math>n\equiv 2t \pmod{4t}</math>
+: Thus, when <math>b = s^2\cdot t</math> with square-free <math>t</math>, and <math>n</math> is [[modular arithmetic|congruent]] to <math>t</math> modulo <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), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the [[Cunningham project]] bases as a product of ''F'', ''L'' and ''M'':<ref>{{cite web|title=Main Cunningham Tables|url=https://homes.cerias.purdue.edu/~ssw/cun/pmain524.txt}} At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ are formulae detailing the aurifeuillean factorizations.</ref>
-: Thus, when <math>b = s^2\cdot t</math> with [[Square-free integer|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 [[Congruence relation|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'':<ref>{{cite web|title=Main Cunningham Tables|url=http://homes.cerias.purdue.edu/~ssw/cun/pmain1215|accessdate=21 December 2015}} 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 998, see <ref>[http://myfactorcollection.mooo.com:8090/LCD_2_199 Coefficients of Lucas C,D polynomials for all square-free bases up to 199]</ref><ref>[myfactorcollection.mooo.com:8090/LCD_2_998 Coefficients of Lucas C,D polynomials for all square-free bases up to 998]</ref>)
+: If we let ''L'' = ''C'' − ''D'', ''M'' = ''C'' + ''D'', the aurifeuillean factorizations for ''b''<sup>''n''</sup> ± 1 of the form ''F'' * (''C'' − ''D'') * (''C'' + ''D'') = ''F'' * ''L'' * ''M'' 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 [[coefficient]]s of the [[polynomial]]s for all square-free bases up to 199 and up to 998, see <ref>[https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean List of aurifeuillean factorization of cyclotomic numbers (square-free bases up to 199)]</ref><ref>[http://myfactorcollection.mooo.com:8090/LCD_2_199 Coefficients of Lucas C,D polynomials for all square-free bases up to 199]</ref><ref>[http://myfactorcollection.mooo.com:8090/LCD_2_998 Coefficients of Lucas C,D polynomials for all square-free bases up to 998]</ref>)
-: (Number = ''F'' * (''C'' − ''D'') * (''C'' + ''D'') = ''F'' * ''L'' * ''M'')
 : {|class="wikitable" style="text-align:center"
@@ Line 160: / Line 157: @@
 |}
+* [[Lucas number]]s <math>L_{10k+5}</math> have the following aurifeuillean factorization:<ref>[https://t5k.org/top20/page.php?id=21 Lucas Aurifeuilliean primitive part]</ref>
-: (See <ref>[http://stdkmd.com/nrr/repunit/repunitnote.htm#aurifeuillean List of Aurifeuillean factorization]</ref> for more information (square-free bases up to 199))
-* 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>
-* [[Lucas number]]s <math>L_{10k+5}</math> have the following aurifeuillean factorization:<ref>[https://web.archive.org/web/20081122085141/http://www.utm.edu/research/primes/lists/top20/LucasA.html Lucas Aurifeuilliean primitive part]</ref>
 :: <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, and <math>F_n</math> is the <math>n</math>th [[Fibonacci number]].
 == History ==
+In 1869, before the discovery of aurifeuillean factorizations, {{ill|Fortuné Landry|lt=Landry|fr||es||de}}, through a tremendous manual effort,<ref name="numericana">[http://www.numericana.com/answer/numbers.htm#aurifeuille Integer Arithmetic, Number Theory – Aurifeuillean Factorizations], Numericana</ref><ref name="primepages">[https://primes.utm.edu/glossary/page.php?sort=GaussianMersenne Gaussian Mersenne], the [[Prime Pages]] glossary</ref> obtained the following factorization into [[prime number|primes]]:
+:<math>2^{58}+1 = 5 \cdot 107367629 \cdot 536903681.</math>
+Three years later, in 1871, [[Léon-François-Antoine Aurifeuille|Aurifeuille]] discovered the nature of this factorization; the number <math>2^{4k+2}+1</math> for <math>k=14</math>, with the formula from the previous section, factors as:<ref name="Wolfram" /><ref name="numericana" />
+:<math>2^{58}+1 = (2^{29}-2^{15}+1)(2^{29}+2^{15}+1) = 536838145 \cdot 536903681.</math>
-In 1871, Aurifeuille discovered the factorization of <math>2^{4k+2}+1</math> for ''k''&nbsp;=&nbsp;14 as the following:<ref name="Wolfram" /><ref name="numericana">[http://www.numericana.com/answer/numbers.htm#aurifeuille Integer Arithmetic, Number Theory – Aurifeuillian Factorizations], Numericana</ref>
+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 [[Édouard Lucas|Lucas]].<ref name="Wolfram" />
-:<math>2^{58}+1 = 536838145 \cdot 536903681.</math>
+536903681 is an example of a [[Mersenne prime#Gaussian Mersenne primes|Gaussian Mersenne]] norm.<ref name="primepages" />
-The second factor is prime, and the factorization of the first factor is <math>5 \cdot 107367629</math>.<ref name="numericana" /> The general form of the factorization was later discovered by [[Édouard Lucas|Lucas]].<ref name="Wolfram" />
 == References ==
@@ Line 182: / Line 179: @@
 == External links ==
-* [http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm Aurifeuillian Factorisation], Colin Barker
+* [http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm Aurifeuillean Factorisation], Colin Barker
+* [http://www.numericana.com/answer/numbers.htm#aurifeuille Aurifeuillean Factorizations], Gérard P. Michon
+* [https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf The Search for Aurifeuillean-Like Factorizations]
 * [http://myfactors.mooo.com/ Online factor collection]
+* [https://www.unshlump.com/hcn/aurif.html A Note on Aurifeuillean Factorizations]
+* [http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm Aurifeuillean Factorisation]
 [[Category:Number theory]]
+[[Category:factorization]]