OFFSET
1,1
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 16167 numbers n such that n^2-1 has no prime factor greater than 97)
PROG
(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 */
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 n^2+1). - Tomohiro Yamada, Apr 22 2017
KEYWORD
nice,nonn,hard
AUTHOR
Charles R Greathouse IV, Jul 23 2010
EXTENSIONS
More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012
STATUS
approved