A244019 Primes of the form 9x^2 + 6xy + 1849y^2.

1873, 2017, 2137, 2377, 2473, 2689, 3217, 3529, 3697, 4057, 4657, 5569, 6073, 6337, 7177, 7393, 7417, 7561, 7681, 7753, 8017, 8089, 8233, 8353, 8737, 8761, 9241, 9601, 9769, 11113, 11257, 11617, 12049, 12433, 12457, 12721, 13297, 13633, 13729, 14281, 15073, 15313, 16417, 17977, 19009, 19273, 20161, 21169, 23017, 24049, 25873, 26161, 26497, 26713, 29569, 30097

Offset: 1

Author

N. J. A. Sloane, Jun 19 2014

Keywords

• nonn

Comments

Discriminant=-66528.

More than the usual number of terms are shown in order to display the difference from A139668 (Primes of the form x^2+1848y^2). The two sequences agree for the first 43 primes but then disagree [Jagy and Kaplansky].

This is a proper subsequence of A139668, since the terms of A244019 have the form z^2 + 1848*y^2: in fact, 9*x^2 + 6*x*y + 1849*y^2 = (3*x+y)^2 + 1848*y^2. [Bruno Berselli, Jun 20 2014]

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Different from A139668 (Primes of the form x^2+1848y^2).

Programs

Maple

fd:=proc(a,b,c,M) local dd,xlim,ylim,x,y,t1,t2,t3,t4,i;
dd:=4*a*c-b^2;
if dd<=0 then error "Form should be positive definite."; break; fi;
t1:={};
xlim:=ceil( sqrt(M/a)*(1+abs(b)/sqrt(dd)));
ylim:=ceil( 2*sqrt(a*M/dd));
for x from 0 to xlim do
for y from -ylim to ylim do
t2 := a*x^2+b*x*y+c*y^2;
if t2 <= M then t1:={op(t1),t2}; fi; od: od:
t3:=sort(convert(t1,list));
t4:=[];
for i from 1 to nops(t3) do
if isprime(t3[i]) then t4:=[op(t4),t3[i]]; fi; od:
[[seq(t3[i],i=1..nops(t3))], [seq(t4[i],i=1..nops(t4))]];
end;
fd(9,6,1849,50000);

Mathematica

Reap[For[p = 2, p < 40000, p = NextPrime[p], s = Solve[x > 0 && 9 x^2 + 6 x y + 1849 y^2 == p, {x, y}, Integers]; If[s != {}, Print[p, " ", {x, y} /. s]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 29 2020 *)