cp's OEIS Frontend

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

%I A175607 #45 Jan 03 2025 14:55:55
%S A175607 3,17,161,8749,19601,246401,672281,23718421,10285001,354365441,
%T A175607 3222617399,9447152318,127855050751,842277599279,2218993446251,
%U A175607 2907159732049,41257182408961,63774701665793,25640240468751,238178082107393,4573663454608289,19182937474703818751,34903240221563713,332110803172167361,99913980938200001
%N A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).
%C A175607 For any prime p, there are finitely many k such that k^2-1 has p as its largest prime factor.
%C A175607 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.
%C A175607 As mentioned by Luca and Najman, this problem is closely related to the one in A002071.
%C A175607 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
%H A175607 Charles R Greathouse IV, <a href="/A175607/b175607.txt">Table of n, a(n) for n = 1..25</a>
%H A175607 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 32.4, pp.632-633.
%H A175607 Florian Luca and Filip Najman, "<a href="http://arxiv.org/abs/1005.1533">On the largest prime factor of x^2-1</a>", Mathematics of Computation 80:273 (2011), pp. 429-435. (Paper has errata that was posted on the MOC website.)
%H A175607 Filip Najman, <a href="http://web.math.hr/~fnajman/">Home Page</a> (gives all 16223 numbers k such that k^2-1 has no prime factor greater than 97)
%o A175607 (PARI)  /* up to term for p=97 */
%o A175607 /* S[] is the list computed by Filip Najman (16223 elements) */
%o A175607 S=[2,3,4, ... ,332110803172167361, 19182937474703818751];
%o A175607 lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
%o A175607 { forprime (p=2, 97,
%o A175607   t = 0;
%o A175607   for (n=1,#S, if ( lpf(S[n]^2-1)==p, t=n ) );
%o A175607   print1(S[t],", ");
%o A175607 );}
%o A175607 /* _Joerg Arndt_, Jul 03 2012 */
%Y A175607 Cf. A214093 (largest primes p such that the greatest prime factor of p^2-1 is prime(n)).
%Y A175607 Cf. A076605 (largest prime divisor of n^2-1).
%Y A175607 Cf. A285283 (equivalent for k^2+1). - _Tomohiro Yamada_, Apr 22 2017
%Y A175607 Cf. A006530, A005563. - _M. F. Hasler_, Jun 13 2018
%K A175607 nonn,nice,hard
%O A175607 1,1
%A A175607 _Charles R Greathouse IV_, Jul 23 2010
%E A175607 More terms (using Filip Najman's list) by _Joerg Arndt_, Jul 03 2012