A185389 Largest number k such that the greatest prime factor of k^2+1 is A002313(n), the n-th prime not congruent to 3 mod 4.

1, 7, 239, 268, 307, 18543, 2943, 485298, 330182, 478707, 24208144, 22709274, 2189376182, 284862638, 599832943, 19696179, 314198789, 3558066693, 69971515635443, 18986886768, 18710140581, 104279454193

Offset: 1

Author

Charles R Greathouse IV, Feb 21 2011

Keywords

• nonn
• hard
• more

Comments

For any prime p, there are finitely many k such that k^2+1 has p as its largest prime factor.

Numbers k such that k^2+1 is p-smooth appear in arctan-relations for the computation of Pi (for example, Machin's identity Pi/4 = 4*arctan(1/5) - arctan(1/239)), see the fxtbook link. [Joerg Arndt, Jul 02 2012]

Links

• Joerg Arndt, Matters Computational (The Fxtbook), section 32.5 "Arctangent relations for Pi", pp. 633-640.
• Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
• 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.
• Filip Najman, Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197)

Crossrefs

Equivalents for other polynomials: A175607 (k^2 - 1), A145606 (k^2 + k).