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%I #14 Apr 15 2017 09:44:15
%S 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,
%T 31,5,31,17,11,17,37,19,37,19,41,7,43,11,43,23,47,23,47,5,17,13,53,13,
%U 53,7,19,29,59,29,61,31,61,31,13,11,67,17,67,17,71,7
%N Largest prime divisor of n^2 - 1.
%C Also the largest prime that divides either n-1 or n+1.
%C 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
%D K. Mahler, "Uber den grossten Primteiler spezieller Polynome zweiten Grades", Arch. Math. Naturvid. B.41, 1935, pp. 3 - 26.
%H T. D. Noe, <a href="/A076605/b076605.txt">Table of n, a(n) for n = 2..10000</a>
%H D. H. Lehmer, <a href="http://projecteuclid.org/euclid.ijm/1256067456">On a problem of Størmer</a>, Ill. J. Math., 8 (1964), 57-79.
%H A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa13113.pdf">On two theorems of Gelfond and some of their applications</a>, Acta Arith. 13 (1967), 177--236.
%H Carl Størmer, <a href="http://www.archive.org/stream/skrifterudgivnea1897chri#page/n79/mode/2up">Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications</a> (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse, 2 (1897), 48 pp.
%e n=11: the largest prime factor of 10 and 12 is 5, therefore a(11) = 5.
%t Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 80}]
%o (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])])))
%Y Cf. A006530, A037464, A074399 (bisections).
%Y Cf. A175607.
%Y Cf. A014442 (largest prime divisor of n^2 + 1). - _Tomohiro Yamada_, Apr 15 2017
%K nonn
%O 2,1
%A _Jon Perry_, Oct 21 2002
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