A141158 - OEIS

%I #58 May 09 2021 07:36:17

%S 5,11,19,29,31,41,59,61,71,79,89,101,109,131,139,149,151,179,181,191,

%T 199,211,229,239,241,251,269,271,281,311,331,349,359,379,389,401,409,

%U 419,421,431,439,449,461,479,491,499,509,521,541,569,571,599,601,619

%N Duplicate of A038872.

%C Original name was: Primes of the form x^2 + 4*x*y - y^2.

%C 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).

%C Values of the quadratic form are {0, 1, 4} mod 5, so this is a subsequence of A038872. - _R. J. Mathar_, Jul 30 2008

%C Is this the same sequence as A038872? [Yes. See a comment in A038872, and the comment by _Jianing Song_ below. - _Wolfdieter Lang_, Jun 19 2019]

%C 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

%C From _Jianing Song_, Sep 20 2018: (Start)

%C 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.

%C (i) If u, v are both even, then choose u = a/2, v = b/2.

%C (ii) If u, v are both odd, 4 | (a-b), then choose u = (3*a+5*b)/4, v = (3*b+a)/4.

%C (iii) If u, v are both odd, 4 | (a+b), then choose u = (3*a-5*b)/4, v = (3*b-a)/4.

%C 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.

%C 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]

%e a(3) = 19 because we can write 19 = 2^2 + 4*2*5 - 5^2.

%t 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 *)

%K dead

%O 1,1

%A 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