A159351 Smallest prime of the form a^2 + b^2 with 0 < a < b such that a + b = 2n+1.

5, 13, 29, 41, 61, 89, 113, 149, 181, 233, 269, 313, 389, 421, 521, 557, 613, 709, 761, 853, 929, 1013, 1109, 1201, 1301, 1409, 1553, 1637, 1741, 1861, 1997, 2113, 2269, 2381, 2521, 2677, 2837, 2969, 3121, 3461, 3449, 3613, 3797, 4001, 4153, 4337, 4513, 4729, 5081

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2009

Keywords

• nonn

Comments

Conjecture: there is always such a prime number.

Primes of the form x^2 + y^2 with 0 < x < y such that there are no primes of the form (x+z)^2 + (y-z)^2 for 0 < z < (y-x)/2. Note: a(40) = 3461 > a(41) = 3449, so the order is not maintained. - Thomas Ordowski, Jan 21 2017

Examples

			1^2 + 2^2 = 5 = a(1) = 1.
2^2 + 3^2 = 13 = a(2) < 1^2 + 4^2 = 17.
2^2 + 5^2 = 29 = a(3) < 1^2 + 6^2 = 37.
23^2 + 32^2 = 1553 = a(27) < 1597, 1657, 1693, 1733, 1777, 1877, 1933, 1993, 2273, 2437, 2617, 2713, 2917, 14 prime representations as sum of two squares.
		

References

• L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
• R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
• David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

Crossrefs

Cf. A145354, A159296, A157884, A264904.

Programs

PARI

isok(p, n) = for (i=1, 2*n, if (i^2 + (2*n+1-i)^2 == p, return (1));); 0;
a(n) = {my(p = 2); while (! isok(p, n), p = nextprime(p+1)); p;} \\ Michel Marcus, Jan 29 2017

Extensions

A-number in definition and cross-reference corrected, and more terms from R. J. Mathar, Apr 24 2009

Edited by Thomas Ordowski, Jan 25 2017