A139668 Primes of the form x^2 + 1848*y^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, 16633, 16657, 16921, 16993, 17257, 17977, 18313, 18481, 19009, 19273, 19441, 20113

Offset: 1

Author

T. D. Noe, Apr 29 2008

Keywords

• nonn
• easy

Comments

Discriminant = -7392.

The primes are congruent to {1, 25, 169, 289, 361, 529, 625, 697, 793, 841, 961, 1345, 1369, 1633, 1681} (mod 1848).

More than the usual number of terms are shown in order to display the difference from A244019. - N. J. A. Sloane, Jun 19 2014

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi).
• 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

Cf. A244019 (a different sequence which agrees for the first 43 terms), A106856.

Programs

Magma

[ p: p in PrimesUpTo(15000) | p mod 1848 in {1, 25, 169, 289, 361, 529, 625, 697, 793, 841, 961, 1345, 1369, 1633, 1681}]; // Vincenzo Librandi, Jul 29 2012

Magma

k:=1848; [p: p in PrimesUpTo(21000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

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(1,0,1848,50000); # N. J. A. Sloane, Jun 19 2014

Mathematica

QuadPrimes2[1, 0, 1848, 10000] (* see A106856 *)