A038873 -id:A038873 - cp's OEIS frontend

A141192 Primes of the form 3x^2+3xy-4y^2 (as well as of the form 8x^2+11xy+2y^2).

2, 3, 29, 41, 53, 59, 71, 89, 107, 113, 167, 173, 179, 227, 257, 269, 281, 293, 317, 383, 401, 431, 449, 509, 521, 563, 569, 599, 641, 659, 677, 683, 743, 773, 797, 827, 839, 857, 863, 887, 911, 941, 953, 971, 977, 983, 1019, 1091, 1097, 1181, 1193, 1229, 1283, 1307, 1319

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 57. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

p = 3 and primes p = 2 mod 3 such that 57 is a square mod p. - Juan Arias-de-Reyna, Mar 20 2011

			a(6)=59 because we can write 59=3*7^2+3*7*8-4*8^2 (or 59=8*1^2+11*1*3+2*3^2)

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

Juan Arias-de-Reyna, 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)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141193 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243192.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 57], Alternatives[2, 8, 14, 29, 32, 41, 50, 53, 56]]&] (* Jean-François Alcover, Oct 28 2016 *)

A141193 Primes of the form -3x^2+3x*y+4y^2 (as well as of the form 6x^2+9xy+y^2).

Original entry on oeis.org

7, 19, 43, 61, 73, 139, 157, 163, 199, 229, 271, 277, 283, 313, 349, 367, 397, 457, 463, 499, 541, 571, 577, 613, 619, 631, 643, 691, 709, 727, 733, 739, 757, 769, 823, 853, 859, 883, 919, 937, 967, 997, 1033, 1051, 1069, 1087, 1201, 1213, 1279, 1297, 1303, 1327, 1423, 1429

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008

nonn

Discriminant = 57. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

p = 19 and primes p = 1 mod 3 such that 57 is a square mod p. - Juan Arias-de-Reyna, Mar 20 2011

			a(2)=19 because we can write 19=-3*1^2+3*1*2+4*2^2

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

Juan Arias-de-Reyna, 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)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141192 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243193.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 19 || MatchQ[Mod[#, 57], Alternatives[1, 4, 7, 16, 25, 28, 43, 49, 55]]&] (* Jean-François Alcover, Oct 28 2016 *)

A141785 Primes of the form -x^2 + 5xy + 5y^2 (as well as of the form 9x^2 + 15xy + 5*y^2).

Original entry on oeis.org

5, 11, 29, 41, 59, 71, 89, 101, 131, 149, 179, 191, 239, 251, 269, 281, 311, 359, 389, 401, 419, 431, 449, 461, 479, 491, 509, 521, 569, 599, 641, 659, 701, 719, 761, 809, 821, 839, 881, 911, 929, 941, 971, 1019, 1031, 1049, 1061, 1091, 1109, 1151, 1181, 1229, 1259

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 45. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(2) = 29 because we can write 29 = -1^2 + 5*1*2 + 5*2^2 (or 29 = 9*1^2 + 15*1*1 + 5*1^2)

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

Juan Arias-de-Reyna, 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. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A033212 (d=45), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 5 || MatchQ[Mod[#, 45], Alternatives[11, 14, 26, 29, 41, 44]]&] (* Jean-François Alcover, Oct 28 2016 *)

A141178 Primes of the form 3x^2+xy-3y^2 (as well as of the form 3x^2+7xy+y^2).

Original entry on oeis.org

3, 7, 11, 37, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149, 151, 157, 173, 181, 197, 211, 223, 229, 233, 263, 269, 271, 293, 307, 317, 337, 349, 359, 367, 373, 379, 397, 419, 433, 443, 491, 509, 521, 571, 593, 599, 601, 613, 617, 619, 641, 659, 673

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 37. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

			a(3) = 11 because we can write 11 = 3*2^2+2*1-3*1^2 (or 11 = 3*1^2+7*1*1+1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Jean-François Alcover, 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)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer-Verlag Berlin Heidelberg, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Primes in A035267.
A subsequence of (and may possibly coincide with) A038913. - R. J. Mathar, Jul 22 2008
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 3*x^2 + x*y - 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 37 || MatchQ[Mod[#, 37], Alternatives[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]]&](* Jean-François Alcover, Oct 25 2016, updated Oct 30 2016 *)

A141191 Primes of the form -2x^2+4xy+5y^2 (as well as of the form 10x^2+16xy+5y^2).

Original entry on oeis.org

5, 7, 13, 31, 47, 61, 101, 103, 157, 167, 173, 181, 199, 223, 229, 269, 271, 293, 311, 349, 367, 383, 397, 439, 461, 479, 503, 509, 607, 647, 661, 677, 719, 727, 733, 773, 797, 829, 839, 853, 887, 941, 983, 997, 1013, 1021, 1039, 1063, 1069, 1109, 1151, 1181

Offset: 1

Laura Caballero Fernandez,Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

nonn

Discriminant = 56. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

Also primes of the form -x^2+6xy+5y^2. cf. A243187.

			a(4)=31 because we can write 31=-2*7^2+4*7*3+5*3^2 (or 31=10*1^2+16*1*1+5*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A141190 (d=56) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Cf. A243187, A243186, A141190.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -2*x^2 + 4*x*y + 5*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

More terms from Colin Barker, Apr 05 2015

A141215 Primes of the form 3x^2+5x*y-3y^2 (as well as 5x^2+9xy+y^2).

Original entry on oeis.org

3, 5, 13, 19, 41, 47, 61, 73, 83, 97, 103, 107, 109, 113, 127, 131, 137, 149, 163, 167, 179, 197, 199, 229, 239, 241, 257, 263, 269, 271, 283, 293, 317, 347, 353, 367, 379, 431, 439, 443, 449, 461, 463, 479, 487, 491, 503, 563, 569, 571, 601, 607, 613, 619

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 14 2008

nonn

Discriminant = 61. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
A subsequence of (and may possibly coincide with) A038941. - R. J. Mathar, Jul 22 2008
3*x^2+5*x*y-3*y^2 and 5*x^2+9*x*y+y^2 are equivalent forms.
Also, primes of the form x^2 - 61y^2, of discriminant 244.

			a(8) = 73 because we can write 73 = 3*4^2+5*4*5-3*5^2 (or 73 = 5*3^2+9*3*1+1^2).

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

Robert Israel, 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. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243654.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

select(p -> isprime(p) and nops([isolve(x^2 - 61*y^2 = p)])>0, [seq(2*i+1,i=1..1000)]); # Robert Israel, Jun 11 2014

terms = 100; d = 61;
Table[3*x^2 + 5*x*y - 3*y^2, {x, 1, terms}, {y, Floor[(5 - Sqrt[d])*x/6], Ceiling[(5 + Sqrt[d])*x/6]}] // Flatten // Select[#, Positive[#] && PrimeQ[#]&]& // Union // Take[#, terms]& (* Jean-François Alcover, Feb 28 2019 *)

A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

Original entry on oeis.org

0, 7, 7, 23, 17, 47, 31, 79, 0, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 0, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 0, 47, 73, 881, 1847, 967, 0, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 0, 3023, 1567, 191, 0, 71

Offset: 2

Klaus Brockhaus, Feb 21 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.

			a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A038873, A059770, A059771, A060515.

f:= proc(n) local P;
  P:= select(`>`,numtheory:-factorset(n^2-2),n);
  if P = {} then 0 else min(P) fi
end proc:
map(f, [$2..100]); # Robert Israel, Feb 23 2016

a[n_] := Module[{P}, P = Select[FactorInteger[n^2 - 2][[All, 1]], # > n&]; If[P == {}, 0, Min[P]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 10 2019, from Maple *)

If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.

Offset corrected by R. J. Mathar, Aug 21 2009

A141160 Primes of the form -x^2 + 3xy + 3y^2 (as well as of the form 5x^2 + 9xy + 3*y^2).

Original entry on oeis.org

3, 5, 17, 41, 47, 59, 83, 89, 101, 131, 167, 173, 227, 251, 257, 269, 293, 311, 353, 383, 419, 461, 467, 479, 503, 509, 521, 563, 587, 593, 647, 677, 719, 761, 773, 797, 839, 857, 881, 887, 929, 941, 971, 983, 1013, 1049, 1091, 1097, 1109, 1151, 1181, 1193

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 21. Class number = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a,b,c) = 1 (primitive).
Except a(1) = 3, primes congruent to {5, 17, 20} mod 21. - Vincenzo Librandi, Jul 11 2018
The comment above is true since the binary quadratic forms with discriminant 21 are in two classes as well as two genera, so there is one class in each genus. A141159 is in the other genus, with primes = 7 or congruent to {1, 4, 16} mod 21. - Jianing Song, Jul 12 2018
4*a(n) can be written in the form 21*w^2 - z^2. - Bruno Berselli, Jul 13 2018
Both forms [-1, 3, 3] (reduced) and [5, 9, 3] (not reduced) are properly (via a determinant +1 matrix) equivalent to the reduced form [3, 3, -1], a member of the 2-cycle [[3, 3, -1], [-1, 3, 3]]. The other reduced form is the principal form [1, 3, -3], with 2-cycle [[1, 3, -3], [-3, 3, 1]] (see, e.g., A141159, A139492). - Wolfdieter Lang, Jun 24 2019

			a(3)=17 because we can write 17 = -1^2 + 3*1*2 + 3*2^2 (or 17 = 5*1^2 + 9*1*1 + 3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A141159, A139492 (d=21) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Primes in A237351.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[3] cat [p: p in PrimesUpTo(2000) | p mod 21 in [5, 17, 20]]; // Vincenzo Librandi, Jul 11 2018

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 3*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)
Join[{3}, Select[Prime[Range[250]], MemberQ[{5, 17, 20}, Mod[#, 21]] &]] (* Vincenzo Librandi, Jul 11 2018 *)

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([-1, 3, 3])
Q.represented_positives(1200, 'prime') # Peter Luschny, Jun 24 2019

More terms from Colin Barker, Apr 05 2015

A141181 Primes of the form 2x^2+3x*y-4y^2 (as well as of the form 2x^2+7xy+y^2).

Original entry on oeis.org

2, 5, 23, 31, 37, 41, 43, 59, 61, 73, 83, 103, 107, 113, 127, 131, 139, 163, 173, 197, 223, 241, 251, 269, 271, 277, 283, 307, 337, 349, 353, 359, 367, 373, 379, 389, 401, 409, 419, 431, 433, 443, 449, 461, 467, 487, 491, 523, 541, 569, 599, 607, 613, 617, 619

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 41. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

It appears that this is the same as "Primes that are squares (mod 41)", cf. A038919 and A373751. - M. F. Hasler, Jun 29 2024

			a(3) = 23 because we can write 23 = 2*3^2+3*3*1-4*1^2 (or 23 = 2*2^2+7*2*1+1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A035269.
A subsequence of (and may possibly coincide with) A038919. - R. J. Mathar, Jul 22 2008
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 2*x^2 + 3*x*y - 4*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

select(p->isprime(p)&&qfbsolve(Qfb(1,7,2),p),[1..1500]) \\ This is to provide a generic characteristic function ("is_A141181") as 1st arg of select(), there are other ways to produce the sequence more efficiently. - M. F. Hasler, Jan 15 2016

A141189 Primes of the form x^2+7xy-y^2 (as well as of the form 7x^2+9x*y+y^2).

Original entry on oeis.org

7, 11, 13, 17, 29, 37, 43, 47, 53, 59, 89, 97, 107, 113, 131, 149, 163, 197, 199, 211, 223, 227, 229, 241, 269, 271, 281, 293, 307, 311, 317, 331, 347, 367, 409, 431, 433, 439, 449, 461, 467, 487, 521, 523, 541, 547, 577, 587, 593, 599, 607, 619, 643, 647, 653

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 53. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

A subsequence of (and may possibly coincide with) A038931. - R. J. Mathar, Jul 22 2008

			a(5) = 29 because we can write 29 = 3^2+7*3*1-1^2 (or 29 = 7*1^2+9*1*2+2^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5). A038873 (d=8). A038931, A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243191.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == x^2 + 7*x*y - y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

cp's OEIS Frontend

A141192 Primes of the form 3*x^2+3*x*y-4*y^2 (as well as of the form 8*x^2+11*x*y+2*y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A141193 Primes of the form -3*x^2+3*x*y+4*y^2 (as well as of the form 6*x^2+9*x*y+y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A141785 Primes of the form -x^2 + 5*x*y + 5*y^2 (as well as of the form 9*x^2 + 15*x*y + 5*y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A141178 Primes of the form 3*x^2+x*y-3*y^2 (as well as of the form 3*x^2+7*x*y+y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A141191 Primes of the form -2*x^2+4*x*y+5*y^2 (as well as of the form 10*x^2+16*x*y+5*y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A141215 Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A141192 Primes of the form 3x^2+3xy-4y^2 (as well as of the form 8x^2+11xy+2y^2).

A141193 Primes of the form -3x^2+3x*y+4y^2 (as well as of the form 6x^2+9xy+y^2).

A141785 Primes of the form -x^2 + 5xy + 5y^2 (as well as of the form 9x^2 + 15xy + 5*y^2).

A141178 Primes of the form 3x^2+xy-3y^2 (as well as of the form 3x^2+7xy+y^2).

A141191 Primes of the form -2x^2+4xy+5y^2 (as well as of the form 10x^2+16xy+5y^2).

A141215 Primes of the form 3x^2+5x*y-3y^2 (as well as 5x^2+9xy+y^2).

A141160 Primes of the form -x^2 + 3xy + 3y^2 (as well as of the form 5x^2 + 9xy + 3*y^2).

A141181 Primes of the form 2x^2+3x*y-4y^2 (as well as of the form 2x^2+7xy+y^2).

A141189 Primes of the form x^2+7xy-y^2 (as well as of the form 7x^2+9x*y+y^2).