|
| |
|
|
A139643
|
|
Primes of the form x^2+Ny^2, with N=102.
|
|
52
|
|
|
|
103, 127, 151, 223, 271, 409, 433, 457, 463, 577, 631, 727, 769, 919, 937, 967, 1033, 1039, 1063, 1087, 1249, 1279, 1327, 1447, 1471, 1543, 1657, 1753, 1759, 1777, 1783, 1801, 1879, 1951, 1993, 2089, 2143, 2161, 2287, 2311, 2473, 2503, 2551
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Discriminant=-408. N is an idoneal number (A000926), which means that the quadratic form's genus consists of a single class, which means that the primes of this form are identical to the primes that are congruent to c (mod 4N), where c is a set of numbers less than 4N. The sequence A139642 lists the set c for each idoneal number. That sequence also cross references the sequences for the quadratic forms with N equal to the first 36 idoneal numbers. The remaining quadratic forms are this sequence and the 28 listed in order below. Note that the sequences for N=120 and 240 are the same.
The primes are congruent to {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361} (mod 408).
|
|
|
REFERENCES
|
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol 3, Chelsea, 1923.
|
|
|
LINKS
|
|
|
|
MAPLE
|
C:= [1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361]:
select(isprime, [seq(seq(408*i+j, j=C), i=0..100)]); # Robert Israel, Jul 03 2016
|
|
|
MATHEMATICA
|
nn=102; pMax=10000; Union[Reap[Do[p=x^2+nn*y^2; If[p<=pMax && PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]] (* T. D. Noe, Aug 02 2009 *)
QuadPrimes2[1, 0, 102, 10000] (* see A106856 *)
|
|
|
PROG
|
(Magma) [ p: p in PrimesUpTo(3000) | p mod 408 in {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361}]; // Vincenzo Librandi, Jul 28 2012
(Magma) k:=102; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016
|
|
|
CROSSREFS
|
Cf. A139644, A139645, A139502, A139646, A139647, A139648, A139506, A139649, A139650, A139651, A139652, A139502, A139653-A139668.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
