A106856 -id:A106856 - cp's OEIS frontend

A106872 Primes of the form 2x^2+xy+4y^2.

2, 5, 7, 19, 41, 59, 71, 97, 101, 103, 107, 109, 113, 157, 163, 191, 193, 211, 233, 257, 281, 307, 311, 317, 359, 373, 397, 419, 421, 439, 443, 467, 479, 503, 541, 547, 563, 593, 599, 659, 661, 683, 691, 701, 727, 733, 751, 769, 877, 887, 907, 977, 997, 1033

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-31.

Primes p such that the polynomial x^3-x^2-1 is irreducible over Zp. The polynomial discriminant is also -31. - T. D. Noe, May 13 2005

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 A123064.

Union[QuadPrimes2[2, 1, 4, 10000], QuadPrimes2[2, -1, 4, 10000]] (* see A106856 *)

A106891 Primes of the form x^2+xy+11y^2.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 43, 47, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139, 167, 173, 181, 193, 197, 229, 239, 251, 269, 271, 281, 283, 293, 307, 311, 317, 337, 353, 359, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509, 541, 547, 557, 563

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-43.
Also, primes of the form x^2-xy+11y^2 with x and y nonnegative.
Also, primes which are a square (mod 43). - M. F. Hasler, Jan 15 2016
Also, primes p such that Legendre(-43,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 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A035233. Cf. A106890.

QuadPrimes2[1, -1, 11, 10000] (* see A106856 *)

select(p->issquare(Mod(p,43))&&isprime(p),[1..1500]) \\ M. F. Hasler, Jan 15 2016

New definition from N. J. A. Sloane, Jun 08 2014

A139668 Primes of the form x^2 + 1848*y^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, 16633, 16657, 16921, 16993, 17257, 17977, 18313, 18481, 19009, 19273, 19441, 20113

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant = -7392.
The primes are congruent to {1, 25, 169, 289, 361, 529, 625, 697, 793, 841, 961, 1345, 1369, 1633, 1681} (mod 1848).
More than the usual number of terms are shown in order to display the difference from A244019. - N. J. A. Sloane, Jun 19 2014

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

Cf. A244019 (a different sequence which agrees for the first 43 terms), A106856.

[ p: p in PrimesUpTo(15000) | p mod 1848 in {1, 25, 169, 289, 361, 529, 625, 697, 793, 841, 961, 1345, 1369, 1633, 1681}]; // Vincenzo Librandi, Jul 29 2012

k:=1848; [p: p in PrimesUpTo(21000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

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(1,0,1848,50000); # N. J. A. Sloane, Jun 19 2014

QuadPrimes2[1, 0, 1848, 10000] (* see A106856 *)

A033209 Primes of form x^2 + 11*y^2.

Original entry on oeis.org

11, 47, 53, 103, 163, 199, 257, 269, 311, 397, 401, 419, 421, 499, 587, 599, 617, 683, 757, 773, 863, 883, 907, 911, 929, 991, 1021, 1087, 1109, 1123, 1181, 1237, 1291, 1307, 1367, 1433, 1439, 1543, 1567, 1571, 1609, 1621, 1697, 1699, 1753, 1873, 1907, 2003

Offset: 1

N. J. A. Sloane

Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 zeros. Compare A106279. - T. D. Noe, May 02 2005

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)
Tim Evink, Paul Alexander Helminck, Tribonacci numbers and primes of the form p=x^2+11y^2, arXiv:1801.04605 [math.NT], 2018.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A243651.

QuadPrimes2[1, 0, 11, 10000] (* see A106856 *)

Extended by T. D. Noe, Apr 17 2012

A106904 Primes of the form x^2-xy+13y^2, with x and y nonnegative.

Original entry on oeis.org

13, 19, 43, 67, 103, 127, 151, 157, 223, 229, 271, 307, 331, 349, 373, 409, 421, 433, 457, 463, 523, 577, 613, 631, 661, 727, 733, 739, 757, 769, 829, 859, 883, 919, 937, 967, 1021, 1033, 1039, 1063, 1069, 1087, 1123, 1171, 1237, 1249, 1279, 1291, 1327

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-51.

Also: Primes which are squares (mod 51). Differs from the subsequence A106903 (because x^2+xy+y^2 = (x+y)^2 - (x+y)y + y^2) from a(20) = 463 on, A106903(20) = 523. Terms which are not in A106903 are: 463, 631, 1033, 1039, 1279, 1291,... Up to 1279 these are also in A139643. Cf. also A191034. - 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)

QuadPrimes2[1, -1, 13, 10000] (* see A106856 *)

select(p->issquare(Mod(p,51))&&isprime(p),[1..1500]) \\ See A106903 for alternative code. - M. F. Hasler, Jan 15 2016

A106933 Primes of the form x^2-xy+17y^2, with x and y nonnegative.

Original entry on oeis.org

17, 19, 23, 29, 37, 47, 59, 67, 71, 73, 83, 89, 103, 107, 127, 131, 149, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 227, 241, 257, 263, 269, 277, 283, 293, 307, 317, 349, 359, 389, 397, 419, 421, 431, 439, 449, 457, 461, 467, 479, 491, 509, 523, 557

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-67.

Also, primes p such that Legendre(-67,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 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

QuadPrimes2[1, -1, 17, 10000] (* see A106856 *)

A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

241, 409, 601, 769, 1009, 1129, 1201, 1249, 1321, 1489, 1609, 1801, 2089, 2161, 2281, 2521, 2689, 3001, 3049, 3121, 3169, 3361, 3529, 3769, 3889, 4129, 4201, 4441, 4561, 4729, 4801, 4969, 5209, 5281, 5449, 5521, 5569, 5641, 5689, 5881, 6121, 6361, 6481

Offset: 1

Artur Jasinski, Apr 24 2008

Also primes of the form x^2 + 120y^2. - T. D. Noe, Apr 29 2008
Also primes of the form x^2+240y^2. See A140633. - T. D. Noe, May 19 2008
In base 12, the sequence is 181, 2X1, 421, 541, 701, 7X1, 841, 881, 921, X41, E21, 1061, 1261, 1301, 13X1, 1561, 1681, 18X1, 1921, 1981, 1X01, 1E41, 2061, 2221, 2301, 2481, 2521, 26X1, 2781, 28X1, 2941, 2X61, 3021, 3081, 31X1, 3241, 3281, 3321, 3361, 34X1, 3661, 3821, 3901, where X is 10 and E is 11. Moreover, the discriminant is 340. - Walter Kehowski, Jun 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

[ p: p in PrimesUpTo(7000) | p mod 120 in {1, 49}]; // Vincenzo Librandi, Jul 28 2012

QuadPrimes2[1, 0, 120, 10000] (* see A106856 *)

The primes are congruent to {1, 49} (mod 120). - T. D. Noe, Apr 29 2008

A033221 Primes of form x^2+31*y^2.

Original entry on oeis.org

31, 47, 67, 131, 149, 173, 227, 283, 293, 349, 379, 431, 521, 577, 607, 617, 653, 811, 839, 853, 857, 919, 937, 971, 1031, 1063, 1117, 1187, 1213, 1237, 1259, 1303, 1327, 1451, 1493, 1523, 1559, 1583, 1619, 1663, 1721, 1723, 1741, 1879, 1931, 1973, 1993, 2003, 2017, 2153, 2273, 2333, 2341, 2521, 2531, 2539, 2543, 2609, 2707, 2711, 2713, 2767, 2797

Offset: 1

N. J. A. Sloane

Also primes of the form x^2+xy+8y^2. - N. J. A. Sloane, Jun 02 2014
Also primes of the form x^2-xy+8y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that the polynomial X^3 + X + 1 splits mod p (see Williams and Hudson link). - Robert Israel, Jun 01 2020

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 2000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
K. Williams and R. Hudson, Representation of primes by the principal form of discriminant -D when the classnumber h(-D) is 3, Acta Arithmetica 57.2 (1991): 131-153.

Primes in A243176.

N:= 10000: # for terms <= N
S:= select(isprime,{31,seq(seq(x^2+31*y^2, y=1..floor(sqrt((N-x^2)/31))),
  x=1..floor(sqrt(N)))}):
sort(convert(S,list)); # Robert Israel, Jun 01 2020

QuadPrimes2[1, 0, 31, 10000] (* see A106856 *)

A033227 Primes of form x^2+39*y^2.

Original entry on oeis.org

43, 103, 139, 157, 181, 277, 367, 439, 523, 547, 607, 673, 751, 823, 991, 997, 1039, 1063, 1117, 1153, 1171, 1231, 1249, 1381, 1429, 1453, 1459, 1483, 1693, 1759, 1933, 1951, 1993, 1999, 2011, 2029, 2131

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+10y^2 with x and y nonnegative. - T. D. Noe, May 07 2005

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)

Primes in A243194.

QuadPrimes2[1, 0, 39, 10000] (* see A106856 *)

A033232 Primes of form x^2+47*y^2.

Original entry on oeis.org

47, 83, 191, 197, 269, 439, 487, 523, 619, 761, 823, 907, 947, 977, 1193, 1277, 1319, 1447, 1481, 1499, 1579, 1693, 1709, 1741, 1811, 1861, 1867, 2053, 2213, 2221, 2273, 2339, 2351, 2447, 2539, 2777

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+12y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

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]
Freddy Cachazo, Diagonally Embedded Sets of (Trop^+ G(2,n))'s in Trop G(2,n): Is There a Critical Value of n?, arXiv:2104.10628 [math.CO], 2021.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A243650.

QuadPrimes2[1, 0, 47, 10000] (* see A106856 *)

cp's OEIS Frontend

A106872 Primes of the form 2x^2+xy+4y^2.

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A106891 Primes of the form x^2+xy+11y^2.

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A139668 Primes of the form x^2 + 1848*y^2.

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A033209 Primes of form x^2 + 11*y^2.

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A106904 Primes of the form x^2-xy+13y^2, with x and y nonnegative.

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A106933 Primes of the form x^2-xy+17y^2, with x and y nonnegative.

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A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

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Magma

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A033221 Primes of form x^2+31*y^2.

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