A076605 Largest prime divisor of n^2 - 1.

3, 2, 5, 3, 7, 3, 7, 5, 11, 5, 13, 7, 13, 7, 17, 3, 19, 5, 19, 11, 23, 11, 23, 13, 5, 13, 29, 7, 31, 5, 31, 17, 11, 17, 37, 19, 37, 19, 41, 7, 43, 11, 43, 23, 47, 23, 47, 5, 17, 13, 53, 13, 53, 7, 19, 29, 59, 29, 61, 31, 61, 31, 13, 11, 67, 17, 67, 17, 71, 7

Offset: 2

Author

Jon Perry, Oct 21 2002

Keywords

• nonn

Comments

Also the largest prime that divides either n-1 or n+1.

Størmer shows that a(n) tends to infinity with n. Schinzel shows that lim inf a(n)/log log n >= 2 and, using lower bounds for linear forms of logarithms, this inequality can be generalized for general quadratic polynomials, with 2 replaced by 4/7 for irreducible ones and 2/7 for reducible ones. - Tomohiro Yamada, Apr 15 2017

Examples

			n=11: the largest prime factor of 10 and 12 is 5, therefore a(11) = 5.
		

References

• K. Mahler, "Uber den grossten Primteiler spezieller Polynome zweiten Grades", Arch. Math. Naturvid. B.41, 1935, pp. 3 - 26.

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000
• D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
• A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), 177--236.
• Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse, 2 (1897), 48 pp.

Crossrefs

Cf. A006530, A037464, A074399 (bisections).

Cf. A175607.

Cf. A014442 (largest prime divisor of n^2 + 1). - Tomohiro Yamada, Apr 15 2017

Programs

Mathematica

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 80}]

PARI

for (n=3,100, print1(","max(factor(n-1)[,1][length(factor(n-1)[,1])],factor(n+1)[,1][length(factor(n+1)[,1])])))