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A141174 Duplicate of A007519. 9
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

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Last modified March 27 02:08 EDT 2017. Contains 284143 sequences.