A159082 - OEIS

%I #15 Jun 14 2022 07:01:51

%S 13,23,29,59,61,73,97,101,103,109,121,127,149,169,187,191,199,221,227,

%T 251,257,263,277,299,307,317,319,331,341,367,373,383,389,397,403,407,

%U 409,433,449,451,461,463,467,491,493,499,517,527,529,533,551,563,571

%N Numbers whose squares added to 7! are prime.

%C 1) Necessarily a(n) is not divisible by 2, 3, 5, 7.

%C 2) Sequence is conjectured to be infinite.

%C 3) It is conjectured that an infinite number of terms are primes.

%C 4) Note that sequence contains a(k), a(k+1) prime twin pairs, first are (59,61), (461,463), (827,829), (1319,1321).

%C 5) It is conjectured that an infinite number of a(n) are squares, first are 121=11^2, 169=13^2, 529=23^2, 841=29^2, 961=31^2, 1681=41^2, ...

%C 6) m!+k^2=n^2 are the generalized Brown number triples (m,k,n).

%D R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, p. 193, 1994

%D I. Niven, H. S. Zuckerman and H. L. Montgomery: An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991

%D David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

%H Harvey P. Dale, <a href="/A159082/b159082.txt">Table of n, a(n) for n = 1..1000</a>

%F 7! + a(n)^2 = prime.

%e 1) 7!+1=71^2, (7, 71) is the largest (of three) Brown pairs; Erdos conjectured that there are no others.

%e 2) 7!+3^2=5049= 3^3 * 11 * 17, 7!+5^2=5065 = 5 * 1013, 7!+7^2=5089 = 7 * 727, 7!+9^2=5121 = 3^2 * 569, 7!+11^2=5161 = 13 * 397.

%e 3) 7!+13^2=5209 prime, so a(1)=13.

%t With[{s = 7!}, Select[Range[600], PrimeQ[#^2 + s] &]] (* _Harvey P. Dale_, Jun 17 2015 *)

%o (PARI) isok(n) = isprime(n^2+7!); \\ _Michel Marcus_, Jul 23 2013; corrected Jun 14 2022

%Y Cf. A038202, A158979.

%K easy,nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 05 2009

%E Edited by _N. J. A. Sloane_, Apr 05 2009