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5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619
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OFFSET
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1,1
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COMMENTS
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Original name was: Primes of the form x^2 + 4*x*y - y^2.
Discriminant = 20. Class number = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a,b,c) = 1 (primitive).
Values of the quadratic form are {0, 1, 4} mod 5, so this is a subsequence of A038872. - R. J. Mathar, Jul 30 2008
Also primes of the form u^2 - 5v^2. The transformation {u,v}={x+2y,y} transforms it into the one in the title. - Tito Piezas III, Dec 28 2008
Yes, this is a duplicate of A038872. For primes p congruent to {1, 4} mod 5, they split in the ring Z[(1+sqrt(5))/2]. Since Z[(1+sqrt(5))/2] is a UFD, they are reducible in Z[(1+sqrt(5))/2], so we have p = e*((a + b*sqrt(5))/2)*((a - b*sqrt(5))/2), where a and b have the same parity and e = +-1. WLOG we can suppose e = 1, otherwise substitute a, b by (a+5*b)/2 and (a+b)/2. Now we show that there exists integer u, v such that p = (u + v*sqrt(5))*(u - v*sqrt(5)) = u^2 - 5*v^2.
(i) If u, v are both even, then choose u = a/2, v = b/2.
(ii) If u, v are both odd, 4 | (a-b), then choose u = (3*a+5*b)/4, v = (3*b+a)/4.
(iii) If u, v are both odd, 4 | (a+b), then choose u = (3*a-5*b)/4, v = (3*b-a)/4.
Hence every prime congruent to {1, 4} mod 5 is of the form u^2 - 5*v^2. On the other hand, u^2 - 5*v^2 == 0, 1, 4 (mod 5). So these two sequences are the same.
Also primes of the form x^2 - x*y - y^2 (discriminant 5) with 0 <= x <= y (or x^2 + x*y - y^2 with x, y nonnegative). (End) [Comment revised by Jianing Song, Feb 24 2021]
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LINKS
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EXAMPLE
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a(3) = 19 because we can write 19 = 2^2 + 4*2*5 - 5^2.
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MATHEMATICA
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lim = 25; Select[Union[Flatten[Table[x^2 + 4 x y - y^2, {x, 0, lim}, {y, 0, lim}]]], # > 0 && # < lim^2 && PrimeQ[#] &] (* T. D. Noe, Aug 31 2012 *)
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CROSSREFS
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KEYWORD
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dead
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AUTHOR
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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
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STATUS
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approved
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