A139668 -id:A139668 - cp's OEIS frontend

A139827 Primes of the form 2x^2 + 2xy + 17y^2.

2, 17, 29, 41, 101, 149, 173, 197, 233, 281, 293, 461, 557, 569, 593, 677, 701, 761, 809, 821, 857, 941, 953, 1097, 1217, 1229, 1289, 1361, 1481, 1493, 1553, 1601, 1613, 1733, 1877, 1889, 1913, 1949, 1997, 2081, 2129, 2141, 2153, 2213, 2273, 2309, 2393, 2417

Offset: 1

T. D. Noe, May 02 2008, May 07 2008

Discriminant = -132.
Consider the quadratic form f(x,y) = ax^2 + bxy + cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in the OEIS.
When a=1 and b=0, f(x,y) is a quadratic form whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such that the Jacobi symbol (-k/4N)=1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A139841-A139843 (d=-408), A139644, A139844-A139850 (d=-420), A139645, A139851-A139853 (d=-448), A139502, A139854-A139860 (d=-480), A139646, A139861-A139863 (d=-520), A139647, A139864-A139866 (d=-532), A139648, A139867-A139873 (d=-660), A139506, A139874-A139880 (d=-672), A139649, A139881-A139883 (d=-708), A139650, A139884-A139886 (d=-760), A139651, A139887-A139893 (d=-840), A139652, A139894-A139896 (d=-928), A139502, A139855, A139857, A139858, A139897-A139899, A139902 (d=-960).
Cf. also  A139653, A139904-A139906 (d=-1012), A139654, A139907-A139913 (d=-1092), A139655, A139914-A139920 (d=-1120), A139656, A139921-A139927 (d=-1248), A139657, A139928-A139934 (d=-1320), A139658, A139935-A139941 (d=-1380), A139659, A139942-A139948 (d=-1428), A139660, A139949-A139955 (d=-1540), A139661, A139956-A139962 (d=-1632), A139662, A139963-A139969 (d=-1848), A139663, A139970-A139976 (d=-2080), A139664, A139977-A139983 (d=-3040), A139665, A139984-A139998 (d=-3360), A139666, A139999-A140013 (d=-5280), A139667, A140014-A140028 (d=-5460), A139668, A140029-A140043 (d=-7392).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[2, -2, 17, 2500] (* see A106856 *)
t = Table[{2, 17, 29, 41, 65, 101} + 132*n, {n, 0, 50}]; Select[Flatten[t], PrimeQ] (* T. D. Noe, Jun 21 2012 *)

v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v,p%132),primes(100)) \\ Charles R Greathouse IV, Jan 08 2013

The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).

A139643 Primes of the form x^2+Ny^2, with N=102.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Apr 29 2008

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).

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.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139644, A139645, A139502, A139646, A139647, A139648, A139506, A139649, A139650, A139651, A139652, A139502, A139653-A139668.

[ 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

k:=102; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

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

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 *)

A139489 Primes of the form x^2+101y^2.

Original entry on oeis.org

101, 137, 677, 1009, 1493, 1693, 1697, 1933, 3137, 3613, 3637, 3701, 3821, 4217, 4261, 4273, 4289, 4373, 4457, 4597, 4861, 5273, 5441, 5849, 6029, 6037, 6473, 6661, 6689, 7193, 7253, 7309, 8377, 8581, 8609, 8677, 9337, 9781, 10133, 10181, 10433

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643-A139668.

See "Binary Quadratic Forms and OEIS" link for list of sequences of primes of the form x^2+ny^2.

a = {}; w = 101; Do[Do[If[PrimeQ[n^2 + w*m^2], AppendTo[a, n^2 + w*m^2]], {n, 1, 700}], {m, 1, 200}]; Union[a]
QuadPrimes2[1,0,101,11000] (* see A106856 *)

101 term prepended by T. D. Noe, Nov 05 2009

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

Original entry on oeis.org

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

N. J. A. Sloane, Jun 19 2014

nonn

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]

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)

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

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);

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 *)

A173274 Primes of the form x^2 + 18480*y^2.

Original entry on oeis.org

18481, 19009, 19441, 20161, 21961, 31249, 41281, 47041, 48409, 51241, 68209, 70009, 70921, 74209, 74449, 74761, 75289, 76129, 76561, 77641, 80809, 84121, 85369, 86689, 87649, 90841, 91081, 91921, 93241, 97441, 102001, 102481, 106681

Offset: 1

Michel Lagneau, Feb 14 2010, Jun 08 2010

nonn

The primes p of the form x^2 + 18480*y^2 are also of the multi-forms x^2 + y^2, x^2 + 2*y^2, x^2 + 3*y^2, ..., x^2 + 11*y^2, x^2 + 12*y^2, but the reverse is false. For example, p = 7561 has twelve forms, but is not of the form x^2 + 18480*y^2.

			18481 = 1^2 + 18480*1^2 and also 18481 = 16^2 + 135^2 = 7^2 + 2*96^2 = 127^2 + 3*28^2 = 135^2 + 4*8^2 = 74^2 + 5*51^2 = 59^2 + 6*50^2 = 97^2 + 7*36^2 = 7^2 + 8*48^2 = 16^2 + 9*45^2 = 29^2 + 10*42^2 = 65^2 + 11*36^2 = 127^2 + 12*14^2.

David A. Cox, "Primes of the Form x^2 + n*y^2", Wiley, 1989, Section 3.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1848, p. 146, Ellipses, Paris 2008.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

Cf. A139668: primes of the form x^2 + 1848*y^2;

Cf. A139665: primes of the form x^2 + 840*y^2.

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(1,0,18480,100000);

QuadPrimes2[1, 0, 18480, 100000] (* see A106856 *)
(* Second program: *)
max = 107000; m = 18480; Table[yy = {y, 1, Floor[Sqrt[max-x^2]/(Sqrt[m])]}; Table[x^2 + m y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]&

fc(a,b,c,M) = {
  my(t1=List(),t2);
  forprime(p=2,prime(M),
    t2 = qfbsolve(Qfb(a,b,c),p);
    if(t2 != 0, listput(t1,p))
  );
  Vec(t1)
};
fc(1,0,18480,100000)

Corrected sequence and replaced defective program. - Ray Chandler, Aug 14 2014

cp's OEIS Frontend

A139827 Primes of the form 2x^2 + 2xy + 17y^2.

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A139643 Primes of the form x^2+Ny^2, with N=102.

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A139489 Primes of the form x^2+101y^2.

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

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A173274 Primes of the form x^2 + 18480*y^2.

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