A285282 - OEIS

%I #19 May 17 2017 05:36:15

%S 1,2,3,5,7,8,18,57,239

%N Numbers n such that n^2 + 1 is 13-smooth.

%C Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 13.

%C Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2*5^a*13^b.

%C This sequence is complete by a theorem of Størmer.

%C The largest instance 239^2 + 1 = 2*13^4 also gives the only nontrivial solution for x^2 + 1 = 2y^4 (Ljunggren).

%D W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = 2y^4, Avh. Norsk Vid. Akad. Oslo. 1(5) (1942), 1--27.

%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 Arithmetica 13 (1967-1968), 177--236.

%H Ray Steiner, <a href="https://doi.org/10.1016/S0022-314X(05)80029-0">Simplifying the Solution of Ljunggren's Equation X^2 + 1 = 2Y^4</a>, J. Number Theory 37 (1991), 123--132, more accesible proof of Ljunggren's result.

%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 Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I Nr. 2 (1897), 48 pp.

%e For n = 8, a(8)^2 + 1 = 57^2 + 1 = 3250 = 2*5^3*13.

%t Select[Range[1000], FactorInteger[#^2 + 1][[-1, 1]] <= 13&] (* _Jean-François Alcover_, May 17 2017 *)

%o (PARI) for(n=1, 9e6, if(vecmax(factor(n^2+1)[, 1])<=13, print1(n", ")))

%o (Python)

%o from sympy import primefactors

%o def ok(n): return max(primefactors(n**2 + 1))<=13 # _Indranil Ghosh_, Apr 16 2017

%Y Cf. A014442, A252493 (n(n+1) instead of n^2 + 1).

%K nonn,fini,full

%O 1,2

%A _Tomohiro Yamada_, Apr 16 2017