A352711 The left Aurifeuillian factor of p^p - 1 for primes p congruent to 1 (mod 4).

11, 1803647, 2699538733, 112663560435723374699, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 133904013361225746608283522164245432446284642589451147, 4429523820749528526448423858097183945539957285504166342434080091097

Offset: 1

Author

Patrick A. Thomas, Mar 30 2022

Keywords

• nonn

Comments

For prime factorizations of p^p - 1 see A125135.

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Nov 04 2022

Examples

			112663560435723374699 is the smaller Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 112663560435723374699.
		

Links

• Patrick A. Thomas, Table of n, a(n) for n = 1..60
• Calculators, Aurifeuillian LMs
• Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
• Wikipedia, Léon-François-Antoine Aurifeuille.
• Wikipedia, Aurifeuillean factorization.

Crossrefs

Cf. A002144, A125135, A230376, A352732, A352400, A352401.

Formula

If R is (p^p-1)/(p-1), where p == 1 (mod 4) and p > 5, then an approximation of the left Aurifeuillian factor of R is (1/e) * sqrt(R/(1+z)), where z =

2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151 or 169 (mod 210),

2/(3p) + 28/(45p^2) + 86/(2835p^3) if p=19,31,61,139,181 or 199 (mod 210),

2/(3p) - 8/(45p^2) + 194/(2835p^3) if p=37,43,67,127,163 or 193 (mod 210),

2/(3p) - 8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157 or 187 (mod 210),

-2/(3p) - 8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),

-2/(3p) - 8/(45p^2) - 194/(2835p^3) if p=17,47,83,143,167 or 173 (mod 210),

-2/(3p) + 28/(45p^2) - 86/(2835p^3) if p=11,29,71,149,179 or 191 (mod 210),

-2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131 or 209 (mod 210).