A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

3, 17, 161, 8749, 19601, 246401, 672281, 23718421, 10285001, 354365441, 3222617399, 9447152318, 127855050751, 842277599279, 2218993446251, 2907159732049, 41257182408961, 63774701665793, 25640240468751, 238178082107393, 4573663454608289, 19182937474703818751, 34903240221563713, 332110803172167361, 99913980938200001

Offset: 1

Author

Charles R Greathouse IV, Jul 23 2010

Keywords

• nonn
• nice
• hard

Comments

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

For every prime p, is there some k where the greatest prime factor of k^2-1 is p? Answer from Artur Jasinski, Oct 22 2010: Yes.

As mentioned by Luca and Najman, this problem is closely related to the one in A002071.

The terms give an upper bound with a method for the simultaneous computation of logarithms of small primes, see the fxtbook link. - Joerg Arndt, Jul 03 2012

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..25
• Joerg Arndt, Matters Computational (The Fxtbook), section 32.4, pp.632-633.
• Florian Luca and Filip Najman, "On the largest prime factor of x^2-1", Mathematics of Computation 80:273 (2011), pp. 429-435. (Paper has errata that was posted on the MOC website.)
• Filip Najman, Home Page (gives all 16223 numbers k such that k^2-1 has no prime factor greater than 97)

Crossrefs

Cf. A214093 (largest primes p such that the greatest prime factor of p^2-1 is prime(n)).

Cf. A076605 (largest prime divisor of n^2-1).

Cf. A285283 (equivalent for k^2+1). - Tomohiro Yamada, Apr 22 2017

Cf. A006530, A005563. - M. F. Hasler, Jun 13 2018

Programs

PARI

/* up to term for p=97 */
/* S[] is the list computed by Filip Najman (16223 elements) */
S=[2,3,4, ... ,332110803172167361, 19182937474703818751];
lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
{ forprime (p=2, 97,
  t = 0;
  for (n=1,#S, if ( lpf(S[n]^2-1)==p, t=n ) );
  print1(S[t],", ");
);}
/* Joerg Arndt, Jul 03 2012 */

Extensions

More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012