A106856 -id:A106856 - cp's OEIS frontend

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

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] See Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107132, A139827, A107003, A141375, A141376 (d=96).
See also A038872 (d=5),
A038873 (d=8),
A068228, A141123 (d=12),
A038883 (d=13),
A038889 (d=17),
A141158 (d=20),
A141159, A141160 (d=21),
A141170, A141171 (d=24),
A141172, A141173 (d=28),
A141174, A141175 (d=32),
A141176, A141177 (d=33),
A141178 (d=37),
A141179, A141180 (d=40),
A141181 (d=41),
A141182, A141183 (d=44),
A033212, A141785 (d=45),
A068228, A141187 (d=48),
A141188 (d=52),
A141189 (d=53),
A141190, A141191 (d=56),
A141192, A141193 (d=57),
A107152, A141302, A141303, A141304 (d=60),
A141215 (d=61),
A141111, A141112 (d=65),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

A106859 Primes of the form 2x^2 + xy + 2y^2.

Original entry on oeis.org

2, 3, 5, 17, 23, 47, 53, 83, 107, 113, 137, 167, 173, 197, 227, 233, 257, 263, 293, 317, 347, 353, 383, 443, 467, 503, 557, 563, 587, 593, 617, 647, 653, 677, 683, 743, 773, 797, 827, 857, 863, 887, 947, 953, 977, 983, 1013, 1097, 1103, 1163, 1187, 1193

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-15.
If p is a prime >= 17 in this sequence then k==0 (mod 4) for all k satisfying "B(2k)(p^k-1) is an integer" where B are the Bernoulli numbers. - Benoit Cloitre, Nov 14 2005
Equals {2, 3, 5 and primes congruent to 17, 23 (mod 30)}; see A039949 and A132235. Except for 2, the same as primes of the form 3x^2 + 5y^2, which has discriminant -60. - T. D. Noe, May 02 2008
Equals {3, 5 and primes congruent to 2, 8 (mod 15)} sorted; see A033212. This form is in the only non-principal class (respectively, genus) for fundamental discriminant -15. - Rick L. Shepherd, Jul 25 2014 [See A343241 for the 2, 8 (mod 15) primes.]
From Wolfdieter Lang, Jun 08 2021: (Start)
Regarding the above comment of T. D. Noe on the form [3, 0, 5]: the class number h(-60) = 2 = A000003(15), and [1, 0, 15] is the principal reduced form, representing the primes given in A033212.
The form [3, 0, 5] represents the proper equivalence class of the second genus of forms of discriminant Disc = -60. The Legendre symbol for the odd primes, not 3 or 5, satisfy L(-3|p) = -1 and L(5|p) = -1, leading to primes p == {17, 23, 47, 53} (mod 60). See the Buell reference, p. 52, for the two characters L(p|3) and L(p|5). The prime 2 is represented by the imprimitive reduced form [2, 2, 8] of Disc = -60. (End)

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 51-52.

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi, next 691 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A000003, A139827, A039949, A132235, A033212, A343241.

QuadPrimes2[2, 1, 2, 100000] (* see A106856 *)

{ fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));
print(t1);
}
fc(2,1,2,1000); \\ N. J. A. Sloane, Jun 06 2014

Removed defective Mma program and extended the b-file using the PARI program fc. - N. J. A. Sloane, Jun 06 2014

A106863 Primes of the form x^2+xy+5y^2.

Original entry on oeis.org

5, 7, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571, 577, 587, 593

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-19.
Also, primes of the form x^2-xy+5y^2 with x and y nonnegative.
Also, primes which are a square (mod 19) (or, (mod 38) - cf. A191028). - M. F. Hasler, Jan 15 2016
Also, primes p such that Legendre(-2,p) = 0 or 1. - N. J. A. Sloane, Dec 25 2017

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 5000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A035243.

QuadPrimes2[1, -1, 5, 10000] (* see A106856 *)

select(p->issquare(Mod(p, 19))&&isprime(p), [1..1000]) \\ M. F. Hasler, Jan 15 2016

A107181 Primes of the form 8x^2 + 9y^2.

Original entry on oeis.org

17, 41, 89, 113, 137, 233, 257, 281, 353, 401, 449, 521, 569, 593, 617, 641, 761, 809, 857, 881, 929, 953, 977, 1049, 1097, 1193, 1217, 1289, 1361, 1409, 1433, 1481, 1553, 1601, 1697, 1721, 1889, 1913, 2081, 2129, 2153, 2273, 2297, 2393, 2417

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -288. See A107132 for more information.
Also primes of the form 9x^2 + 6xy + 17y^2. See A140633. - T. D. Noe, May 19 2008
All terms are of the form x^2 + y^2, see A002144. - Zak Seidov, Jan 26 2014

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)

Subsequence of A002144 (Pythagorean primes).

Cf. A139827.

[ p: p in PrimesUpTo(5000) | p mod 24 eq 17 ]; // Vincenzo Librandi, Apr 19 2011

QuadPrimes2[8, 0, 9, 10000] (* see A106856 *)

list(lim)=my(v=List()); forprime(p=17,lim, if(p%24==17, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

The primes are congruent to 17 (mod 24). - T. D. Noe, May 02 2008

A033207 Primes of the form x^2 + 7*y^2.

Original entry on oeis.org

7, 11, 23, 29, 37, 43, 53, 67, 71, 79, 107, 109, 113, 127, 137, 149, 151, 163, 179, 191, 193, 197, 211, 233, 239, 263, 277, 281, 317, 331, 337, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 457, 463, 487, 491

Offset: 1

N. J. A. Sloane

Except for a(1) = 7, these are the primes which can be written in the form a^2 + 7*b^2 with a > 0 and b > 0. - V. Raman, Sep 08 2012

These are the primes p for which p^3 - 1 is divisible by 7, with two exceptions: p = 2 is not in the sequence even though 2^3 - 1 is divisible by 7, and p = 7 is in the sequence even though 7^3 - 1 is not divisible by 7. Except for p = 7, if p^3 - 1 is not divisible by 7, it is congruent to 5 (mod 7). - Richard R. Forberg, Jun 03 2013

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)
Sushma Palimar and B. R. Shankar, Mersenne Primes in Real Quadratic Fields, Journal of Integer Sequences, Vol. 15 (2012), #12.5.6.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Essentially the same as A045373. Primes in A020670.

Cf. A002344, A002345, A139643.

QuadPrimes2[1, 0, 7, 10000] (* see A106856 *)

is(n)=kronecker(n,7)>=0 && isprime(n) && n>2 \\ Charles R Greathouse IV, Nov 19 2012

Primes congruent to {1, 7, 9, 11, 15, 23, 25} (mod 28). - T. D. Noe, Apr 29 2008

A102271 Primes of the form 3x^2 + 7y^2.

Original entry on oeis.org

3, 7, 19, 31, 103, 139, 199, 223, 271, 283, 307, 367, 439, 523, 607, 619, 643, 691, 727, 787, 811, 859, 1039, 1063, 1123, 1231, 1279, 1291, 1399, 1447, 1459, 1483, 1531, 1543, 1567, 1627, 1699, 1783, 1867, 1879, 1951, 1987, 2131, 2203, 2239, 2287, 2371, 2383

Offset: 1

N. J. A. Sloane, Feb 19 2005

Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = -1.

Ray Chandler, Table of n, a(n) for n = 1..10000 (First 1000 terms from Vincenzo Librandi).
H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A102269-A102275, A139827.

[p: p in PrimesUpTo(3000) | p mod 84 in [3, 7, 19, 31, 55]]; // Vincenzo Librandi, Jul 19 2012

m=3; n=7; pLst={}; lim=3000; xMax=Sqrt[lim/m]; yMax=Sqrt[lim/n]; Do[p=m*x^2+n*y^2; If[pT. D. Noe, May 05 2005 *)
QuadPrimes2[3, 0, 7, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\7), if(isprime(t=w+7*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {3, 7, 19, 31, 55} (mod 84). - T. D. Noe, May 02 2008

A106881 Primes of the form x^2+xy+9y^2.

Original entry on oeis.org

11, 29, 71, 79, 109, 149, 151, 179, 191, 211, 239, 281, 331, 359, 379, 389, 401, 421, 431, 449, 491, 499, 541, 569, 571, 599, 631, 641, 659, 701, 709, 739, 751, 809, 821, 911, 919, 991, 1009, 1019, 1031, 1051, 1061, 1129, 1171, 1201, 1229, 1289, 1381

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-35.
Also, primes of the form x^2-xy+9y^2, with x and y nonnegative.
Also, primes which are squares (mod 35). A subsequence of A191026. - M. F. Hasler, Jan 15 2016

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)

Primes in A243178.

QuadPrimes2[1, -1, 9, 10000] (* see A106856 *)

select(p->issquare(Mod(p,35))&&isprime(p),[1..1000]) \\ M. F. Hasler, Jan 15 2016

A107007 Primes of the form 3x^2+8y^2.

Original entry on oeis.org

3, 11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-96.
Except for 3, also primes of the forms 8*x^2+8*x*y+11*y^2 and 11*x^2+6*x*y+27*y^2. See A140633. - T. D. Noe, May 19 2008
Except for the first member, 3, all the members seem to be terms of A123239 which are prime in both k(i) and k(rho). - A.K. Devaraj, Nov 24 2009

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. A139827.

[3] cat[ p: p in PrimesUpTo(3000) | p mod 24 in {11} ]; // Vincenzo Librandi, Jul 23 2012

QuadPrimes2[3, 0, 8, 10000] (* see A106856 *)

Except for 3, the terms are congruent to 11 (mod 24). - T. D. Noe, May 02 2008

A033217 Primes of form x^2 + 23*y^2.

Original entry on oeis.org

23, 59, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 829, 853, 877, 883, 991, 997, 1097, 1117, 1151, 1163, 1181, 1231, 1319, 1451, 1453, 1481, 1553, 1613, 1669, 1697, 1787, 1789, 1867, 1871, 1879, 1889, 1913, 2027, 2053, 2143, 2309, 2339, 2347, 2381, 2393, 2423, 2539, 2647, 2677, 2693, 2707, 2741, 2819

Offset: 1

N. J. A. Sloane

Discriminant -23.
Also primes of the form x^2 + x*y + 6*y^2. - N. J. A. Sloane, Jun 02 2014
Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that X^3-X+1 is split modulo p. E.g., X^3-X+1 = (X-33)*(X-40)*(X-94) modulo 167. - Julien Freslon (julien.freslon(AT)wanadoo.fr), Feb 24 2007
It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from Jud McCranie
In fact, this sequence appears to be the same as primes p such that RamanujanTau(p) == {1,2} (mod 23). - Ray Chandler, Dec 01 2016
Excluding the first term, this sequence is the intersection of A191021 and A256567. - Arkadiusz Wesolowski, Oct 03 2021
From Amiram Eldar, Jan 10 2025: (Start)
a(2)..a(10000) are the first terms of the sequence of primes p such that tau(p) == 2 (mod 23), where tau is Ramanujan's tau function (A000594).
Moree and Noubissie (2024) proved that the following 3 conditions for a prime p are equivalent:
  1. tau(p) == 2 (mod 23).
  2. p divides A000931(p+3) where A000931 is the Padovan sequence.
  3. The number of distinct roots modulo p of the polynomial x^3 - x - 1 is 3. (End)

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992. See pp. 158-160, "Integer 23 - the Tau function".

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Pieter Moree and Armand Noubissie, Higher reciprocity laws and ternary linear recurrence sequences, International Journal of Number Theory, 2024; arXiv preprint, arXiv:2205.06685 [math.NT], 2022.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.
John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Cf. A000594, A191021, A256567. Primes in A028958.

QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
Join[{23}, nn=23; pMax=5000; 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]]]] (* Vincenzo Librandi, Sep 05 2016 *)

isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ Arkadiusz Wesolowski, Oct 03 2021

isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ Arkadiusz Wesolowski, Oct 09 2021

A084865 Primes of the form 2x^2 + 3y^2.

Original entry on oeis.org

2, 3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 251, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 683, 701, 773, 797, 821, 827, 941, 947, 971, 1013, 1019, 1061, 1091, 1109, 1163, 1181, 1187

Offset: 1

Reinhard Zumkeller, Jun 10 2003

Subsequence of A084864; A084863(a(n))>0.
Conjecture: A084863(a(n))=1?
Is it true that a(n) = A019338(n+1)?
Comment: The truth of the conjecture A084863(a(n))=1 follows from the genus theory of quadratic forms (see Cox, page 61). By comparing enough terms, we see that the conjecture a(n) = A019338(n+1) is false. - T. D. Noe, May 02 2008
Appears to be the primes p such that (p mod 6)*(Fibonacci(p) mod 6)=25. - Gary Detlefs, May 26 2014

			A000040(17) = 59 = 32 + 27 = 2*4^2 + 3*3^2, therefore 59 is a term.

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

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. A084863, A084864, A019338, A084866, A139827.

Primes in A002480.

QuadPrimes2[2, 0, 3, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\3), if(isprime(t=w+3*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {2, 3, 5, 11} (mod 24). - T. D. Noe, May 02 2008

cp's OEIS Frontend

A141373 Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

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A106859 Primes of the form 2x^2 + xy + 2y^2.

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A106863 Primes of the form x^2+xy+5y^2.

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A107181 Primes of the form 8x^2 + 9y^2.

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A033207 Primes of the form x^2 + 7*y^2.

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A102271 Primes of the form 3*x^2 + 7*y^2.

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A106881 Primes of the form x^2+xy+9y^2.

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A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

A102271 Primes of the form 3x^2 + 7y^2.

A107007 Primes of the form 3x^2+8y^2.