OFFSET
1,1
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
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
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2009
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
STATUS
approved