



17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Originally "Primes of the form x^2 + 4xy  4y^2 (as well as of the form x^2 + 6xy + y^2)."
R. J. Mathar was the first to wonder whether these are also primes of the form 8k + 1. I did the easy part, proving that all primes of the form x^2 + 4xy  4y^2 are congruent to 1 mod 8. Since x^2 + 4xy  4y^2 = 2 or 2 is impossible, x must be odd. And since x is odd, x^2 = 1 mod 8.
If y is even, then both 4xy and 4y^2 are multiples of 8. If y is odd, then 4xy = 4 mod 8, but so is 4y^2, cancelling out the effect and leaving x^2 = 1 mod 8.
It remains to prove that every prime of the form 8k + 1 has a representation as x^2 + 4xy  4y^2.  Alonso del Arte, Jan 28 2017
A necessary and sufficient condition of representation of p = 8n + 1 in your quadratic form is {8y^2 + 8n + 1 is perfect square}, since only in this case solving square equation for x, we have x = 2y + sqrt(8y^2 + 8n + 1) is [an] integer. For this a sufficient condition is { n has a form n = k^2  k + i(4k + i  1)/2, i >= 0, k >= 1}. In this case x = 2i + 2k  1. y = k."  Vladimir Shevelev, Jan 26 2017


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000


CROSSREFS

Sequence in context: A263012 A172280 A004625 * A007519 A163185 A138005
Adjacent sequences: A141171 A141172 A141173 * A141175 A141176 A141177


KEYWORD

nonn,dead


AUTHOR

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008


EXTENSIONS

More terms from Michel Marcus, Feb 01 2014


STATUS

approved



