A285282 Numbers n such that n^2 + 1 is 13-smooth.

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

Offset: 1

Author

Tomohiro Yamada, Apr 16 2017

Keywords

• nonn
• fini
• full

Comments

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

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.

This sequence is complete by a theorem of Størmer.

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

Examples

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

References

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

Links

• A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
• Ray Steiner, Simplifying the Solution of Ljunggren's Equation X^2 + 1 = 2Y^4, J. Number Theory 37 (1991), 123--132, more accesible proof of Ljunggren's result.
• Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I Nr. 2 (1897), 48 pp.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Python

from sympy import primefactors
def ok(n): return max(primefactors(n**2 + 1))<=13 # Indranil Ghosh, Apr 16 2017