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Primes of the form x^2 + 45y^2.
+10
37
61, 109, 181, 229, 241, 349, 409, 421, 541, 601, 661, 709, 769, 829, 1009, 1021, 1069, 1129, 1201, 1249, 1321, 1381, 1429, 1489, 1549, 1609, 1621, 1669, 1741, 1789, 1801, 1861, 2029, 2089, 2161, 2221, 2269, 2281, 2341, 2389, 2521, 2689, 2749, 3001, 3049, 3061, 3109, 3121, 3169, 3181
COMMENTS
Discriminant = -180. See A107132 for more information. Also primes of the form x^2+6*x*y-6*y^2, of discriminant 60 (as well as of the form x^2+8*x*y+y^2). - 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
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
FORMULA
Primes congruent to {1, 49} (mod 60). - T. D. Noe, Apr 29 2008
MATHEMATICA
QuadPrimes2[1, 0, 45, 10000] (* see A106856 *)
PROG
(Magma) [ p: p in PrimesUpTo(3000) | p mod 60 in {1, 49 } ]; // Vincenzo Librandi, Jul 24 2012 (PARI) list(lim)=my(v=List(), t); forprime(p=61, lim, t=p%60; if(t==1||t==49, listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017
CROSSREFS
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.
+10
11
6, 11, 14, 15, 24, 35, 44, 51, 54, 56, 59, 60, 71, 86, 96, 99, 110, 119, 126, 131, 134, 135, 140, 150, 159, 176, 179, 191, 204, 206, 215, 216, 224, 231, 236, 239, 240, 251, 254, 275, 284, 294, 311, 315, 326, 335, 339, 344, 350, 359, 366, 371, 374, 375, 384
COMMENTS
Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c]. Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n]. Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) = k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.
Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.
Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.
Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)
EXAMPLE
6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).
Primes of the form 2*x^2+6*x*y-3*y^2 (as well as of the form 5*x^2+10*x*y+2*y^2).
+10
10
2, 5, 17, 53, 113, 137, 173, 197, 233, 257, 293, 317, 353, 557, 593, 617, 653, 677, 773, 797, 857, 953, 977, 1013, 1097, 1193, 1217, 1277, 1373, 1433, 1493, 1553, 1613, 1637, 1697, 1733, 1877, 1913, 1973, 1997, 2153, 2213, 2237, 2273, 2297, 2333, 2357, 2393, 2417, 2477
COMMENTS
Discriminant = 60. Class = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
This is also the list of primes p such that p = 2 or 5 or p is congruent to 17 or 53 mod 60. - Jean-François Alcover, Oct 28 2016
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
a(3)=17 because we can write 17=2*2^2+6*2*1-3*1^2 (or 17=5*1^2+10*1*1+2*1^2).
MATHEMATICA
Select[Prime[Range[500]], # == 2 || # == 5 || MatchQ[Mod[#, 60], 17|53]&] (* Jean-François Alcover, Oct 28 2016 *)
AUTHOR
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
Primes of the form -2*x^2+6*x*y+3*y^2 (as well as of the form 7*x^2+12*x*y+3*y^2).
+10
10
3, 7, 43, 67, 103, 127, 163, 223, 283, 307, 367, 463, 487, 523, 547, 607, 643, 727, 787, 823, 883, 907, 967, 1063, 1087, 1123, 1303, 1327, 1423, 1447, 1483, 1543, 1567, 1627, 1663, 1723, 1747, 1783, 1867, 1987, 2083, 2143, 2203, 2287, 2347, 2383, 2467, 2503, 2647, 2683
COMMENTS
Discriminant = 60. Class = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1
This is also the list of primes p such that p = 3 or p is congruent to 7 or 43 mod 60. - Jean-François Alcover, Oct 28 2016
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
a(3)=43 because we can write 43=-2*1^2+6*1*3+3*3^2 (or 43=7*1^2+12*1*2+3*2^2).
AUTHOR
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
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).
+10
9
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
COMMENTS
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
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
FORMULA
Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008
EXAMPLE
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).
MATHEMATICA
QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)
PROG
(Magma) [3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012 (PARI) 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
AUTHOR
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
EXTENSIONS
Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.
Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.
+10
3
0, 2, 5, 8, 17, 18, 20, 32, 33, 42, 45, 50, 53, 68, 72, 77, 80, 98, 105, 113, 122, 125, 128, 132, 137, 153, 162, 168, 170, 173, 177, 180, 197, 200, 212, 213, 218, 233, 242, 245, 257, 258, 272, 288, 293, 297, 305, 308, 317, 320, 330, 338, 353, 357, 362, 378
COMMENTS
Discriminant 60.
Nonnegative integers k such that 3x^2 - 5y^2 + k = 0 has integer solutions.
Also nonnegative integers of the form 2x^2 + (4m+2)xy + (2m^2+2m-7)y^2 for integers m. This includes the form in the name with m = 1.
Also nonnegative integers of the form 5x^2 + 10mxy + (5m^2-3)y^2 for integers m. This includes the form from Jon E. Schoenfield above with m = 0.
There are no squares in this sequence. Even powers of terms as well as products of an even number of terms belong to A243188. Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the form 5x^2 - 3y^2 by the following identity: (na^2 - kb^2)(nc^2 - kd^2)(ne^2 - kf^2) = n[a(nce + kdf) + bk(cf + de)]^2 - k[na(cf + de) + b(nce + kdf)]^2 for all a, b, c, d, e, f, k, n in R. This can be verified by expanding both sides of the equation.
(End)
MATHEMATICA
Reap[For[n = 0, n <= 200, n++, If[Reduce[2*x^2 + 6*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]
Nonnegative numbers of the form -2x^2+6xy+3y^2.
+10
3
0, 3, 7, 12, 22, 27, 28, 30, 43, 48, 55, 63, 67, 70, 75, 88, 102, 103, 108, 112, 118, 120, 127, 142, 147, 163, 172, 175, 183, 187, 192, 198, 220, 223, 238, 243, 252, 255, 262, 268, 270, 280, 283, 295, 300, 307, 318, 327, 343, 352, 355, 358, 363, 367, 382
COMMENTS
Discriminant 60.
Also: nonnegative 3x^2-5y^2 since 3y^2+6xy-2x^2 = 3(y+x)^2-5x^2. - R. J. Mathar, Jun 10 2020
MATHEMATICA
Reap[For[n = 0, n <= 200, n++, If[Reduce[-2*x^2 + 6*x*y + 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]
Primes of the form 4*x^2 + 3*x*y - 4*y^2 (as well as of the form 2*x^2 + 9*x*y + y^2).
+10
2
2, 3, 19, 23, 37, 41, 61, 67, 71, 73, 79, 89, 97, 109, 127, 137, 149, 173, 181, 211, 223, 227, 251, 257, 269, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 383, 389, 397, 401, 419, 439, 457, 461, 463, 479, 487, 499, 503, 509, 523, 547, 557, 587, 593, 607
COMMENTS
Discriminant = 73. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
a(2) = 3 because we can write 3 = 4*1^2 + 3*1*1 - 4*1^2.
CROSSREFS
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). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).
AUTHOR
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 03 2008
Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2).
+10
2
3, 5, 7, 17, 23, 37, 73, 97, 107, 113, 163, 167, 173, 193, 197, 227, 233, 277, 283, 313, 317, 337, 347, 367, 397, 487, 503, 547, 607, 617, 643, 653, 673, 677, 683, 743, 787, 823, 827, 853, 857, 877, 887, 907, 947, 983, 997, 1013, 1093, 1117, 1153, 1163, 1187
COMMENTS
Discriminant = 85. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
a(1) = 3 because we can write 3 = 3*1^2 + 5*1*0 - 5*0^2 (or 3 = 7*0^2 + 13*0*1 + 3*1^2).
CROSSREFS
Cf. A141773 (d=85). 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). A141750 (d=73). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).
AUTHOR
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008
Primes of the form 4*x^2 + 3*x*y - 5*y^2 (as well as of the form 8*x^2 + 11*x*y + y^2).
+10
1
2, 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139, 157, 167, 173, 179, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 283, 307, 311, 317, 331, 347, 367, 373, 401, 409, 443, 449, 461, 463, 467, 479, 487, 509, 523, 587, 601, 607, 613, 619, 631
COMMENTS
Discriminant = 89. 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.
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
a(1) = 2 because we can write 2 = 4*1^2 + 3*1*1 - 5*1^2.
CROSSREFS
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). A141750 (d=73). A141772, A141773 (d=85). A141776, A141777 (d=88). A141778 (d=89). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).
AUTHOR
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008
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