A159082 Numbers whose squares added to 7! are prime.

13, 23, 29, 59, 61, 73, 97, 101, 103, 109, 121, 127, 149, 169, 187, 191, 199, 221, 227, 251, 257, 263, 277, 299, 307, 317, 319, 331, 341, 367, 373, 383, 389, 397, 403, 407, 409, 433, 449, 451, 461, 463, 467, 491, 493, 499, 517, 527, 529, 533, 551, 563, 571

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

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

2) Sequence is conjectured to be infinite.

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

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

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, ...

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

Examples

			1) 7!+1=71^2, (7, 71) is the largest (of three) Brown pairs; Erdos conjectured that there are no others.
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.
3) 7!+13^2=5209 prime, so a(1)=13.
		

References

• R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, p. 193, 1994
• I. Niven, H. S. Zuckerman and H. L. Montgomery: An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991
• David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A038202, A158979.

Programs

Mathematica

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

PARI

isok(n) = isprime(n^2+7!); \\ Michel Marcus, Jul 23 2013; corrected Jun 14 2022

Formula

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

Extensions

Edited by N. J. A. Sloane, Apr 05 2009