A096172 Largest prime factor of n^4 + 1.

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 137, 7321, 233, 14281, 937, 1489, 65537, 41761, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 1129, 61681, 6113, 1336337, 750313, 98801, 10529, 50857, 1156721

Offset: 1

Author

Hugo Pfoertner, Jun 19 2004

Keywords

• nonn

Comments

Mabkhout shows that a(n) >= 137 for n > 3. - Charles R Greathouse IV, Apr 07 2014

Examples

			a(1)=2 because 1^4 + 1 = 2;
a(2)=17: 2^4 + 1 = 17;
a(8)=241: 8^4 + 1 = 4097 = 17*241.
		

References

• Mustapha Mabkhout, Minoration de P(x^4+1), Rendiconti del Seminario della Facoltà di Scienze dell'Università di Cagliari 63:2 (1993), pp. 135-148.

Links

• Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
• Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
• Marina Mureddu, A lower bound for P(x^4+1), Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 5, Vol. 8, No. 2 (1986-1987), pp. 109-119.

Crossrefs

Cf. A000068, A002523, A014442, A037896, A006530, A069170, A096169, A096171.

Programs

Mathematica

FactorInteger[#^4+1][[-1,1]]&/@Range[40] (* Harvey P. Dale, Apr 30 2012 *)

PARI

a(n)=my(f=factor(n^4+1)[,1]); f[#f] \\ Charles R Greathouse IV, Apr 07 2014

Formula

a(n) = A006530(1+n^4) = A014442(n^2). - R. J. Mathar, Jan 28 2017

From Amiram Eldar, Oct 28 2024: (Start)

a(n) > 113 for n > 3 (Mureddu, 1986-1987).

a(n) >= 233 for n >= 11 (Luca, 2004). (End)