A106856 -id:A106856 - cp's OEIS frontend

A033215 Primes of form x^2+21*y^2.

37, 109, 193, 277, 337, 373, 421, 457, 541, 613, 673, 709, 757, 877, 1009, 1033, 1093, 1117, 1129, 1201, 1213, 1297, 1381, 1429, 1453, 1549, 1597, 1621, 1789, 1801, 1873, 1933, 2017, 2053, 2137, 2221

Offset: 1

N. J. A. Sloane

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. A139643.

QuadPrimes2[1, 0, 21, 10000] (* see A106856 *)

Equivalently, primes congruent to {1, 25, or 37} (mod 84). - T. D. Noe, Apr 29 2008 [See e.g. Cox, p. 36. - N. J. A. Sloane, May 27 2014]

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

A107006 Primes of the form 4x^2-4xy+7y^2, with x and y nonnegative.

Original entry on oeis.org

7, 31, 79, 103, 127, 151, 199, 223, 271, 367, 439, 463, 487, 607, 631, 727, 751, 823, 919, 967, 991, 1039, 1063, 1087, 1231, 1279, 1303, 1327, 1399, 1423, 1447, 1471, 1543, 1567, 1663, 1759, 1783, 1831, 1879, 1951, 1999, 2143, 2239, 2287, 2311

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-96.
Also, primes of the form 24n+7. - Artur Jasinski, Nov 25 2007 [See the Reble link]
Also primes of the forms 4x^2+4xy+7y^2, 7x^2+6xy+15y^2, 7x^2+2xy+7y^2 and 7x^2+4xy+28y^2. See A140633. - T. D. Noe, May 19 2008
Also, primes of form u^2+6v^2 with odd v while sequence A107008 is even v. This can be seen by expressing its form as (2x-y)^2+6y^2 (where y can only be odd) while the latter is x^2+6(2y)^2. Additionally, this sequence is 7 mod 24 while the second is 1 mod 24 and together, they are the primes of form x^2+6y^2 (A033199) which are either {1,7} mod 24. - Tito Piezas III, Jan 01 2009

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 168 terms from N. J. A. Sloane]
Don Reble, Notes on this sequence
J. Liouville, Théorème concernant les nombres premiers de la forme 24µ + 7, Journal de mathématiques pures et appliquées 2e série, tome 4 (1859), pp. 399-400.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A124477.

a = {}; Do[If[PrimeQ[24n + 7], AppendTo[a, 24n + 7]], {n, 0, 100}]; a (* Artur Jasinski, Nov 25 2007 *)
QuadPrimes2[4, -4, 7, 10000] (* see A106856 *)
Select[24*Range[0,4000]+7,PrimeQ] (* Harvey P. Dale, May 13 2018 *)

Recomputed b-file and deleted erroneous Mma program by N. J. A. Sloane, Jun 08 2014

A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

Original entry on oeis.org

3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-2)). - N. J. A. Sloane, Dec 25 2017
Fermat knew of the relationship between a prime being congruent to 1 or 3 mod 8 and its being the sum of a square and twice a square, and claimed to have a firm proof of this fact. These numbers are not primes in Z[sqrt(-2)], as they have x - y sqrt(-2) as a divisor. - Alonso del Arte, Dec 07 2012
Terms m in A047471 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = + 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For the minimal polynomial C see A187360. For p == 1 (mod 8) the norm is C(p, 0) (see a comment on 4*A005123) and for p == 3 (mod 8) the norm is -C(p, 0) (see a comment on A186297). For the primes with norm(rho(p)) = -1 see A003628. - Wolfdieter Lang, Oct 24 2013
If p is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3]. - N. J. A. Sloane, May 30 2014
Primes that are the quarter perimeter of a Heronian triangle. Such primes are unique to the Heronian triangle (see Yiu link). - Frank M Jackson, Nov 30 2014

			Since 11 is prime and 11 == 3 (mod 8), 11 is in the sequence. (Also 11 = 3^2 + 2 * 1^2 = (3 + sqrt(-2))(3 - sqrt(-2)).)
Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe]
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.
Zak Seidov, Table of n, a(n), x and y for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Paul Yiu, CRUX, Problem 2331, Proposed by Paul Yiu
Paul Yiu and Jill S. Taylor, CRUX, Problem 2331, Solution pp 185-186
Index to sequences related to decomposition of primes in quadratic fields

Cf. A033203.

a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012

Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* Harvey P. Dale, Jun 09 2017 *)

is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = A033203(n+1). - Zak Seidov, May 29 2014
A007519 UNION A007520. - R. J. Mathar, Jun 09 2020
L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -Wolfdieter Lang, Jul 24 2024

A033210 Primes of the form x^2+13*y^2.

Original entry on oeis.org

13, 17, 29, 53, 61, 101, 113, 157, 173, 181, 233, 257, 269, 277, 313, 337, 373, 389, 433, 521, 569, 601, 641, 653, 673, 677, 701, 757, 797, 809, 829, 857, 881, 937, 953, 997, 1013, 1049, 1069, 1093, 1109, 1117

Offset: 1

N. J. A. Sloane

First differences are multiples of 4 (which follows from set of differences of the moduli in the Noe formula). Minimal difference 4 occurs at a(1)=17, a(25)=673, a(48)=1297, etc. - Zak Seidov, Oct 04 2014

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 3000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A248212 (x) and A248213 (y).

QuadPrimes2[1, 0, 13, 10000] (* see A106856 *)

select(n->vecsearch([1,9,13,17,25,29,49],n%52), primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

is_A033210(n)={vecsearch([1,9,13,17,25,29,49],n%52)&&isprime(n)} \\ setsearch() is still slower by a factor > 2. - M. F. Hasler, Oct 04 2014

Same as primes congruent to {1, 9, 13, 17, 25, 29, or 49} (mod 52). - T. D. Noe, Apr 29 2008 [See e.g. Cox, p. 36. - N. J. A. Sloane, May 27 2014]

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

A056874 Primes of form x^2+xy+3y^2, discriminant -11.

Original entry on oeis.org

3, 5, 11, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 179, 181, 191, 199, 223, 229, 251, 257, 269, 311, 313, 317, 331, 353, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521, 577, 587, 599

Offset: 1

N. J. A. Sloane, Sep 02 2000

Also, primes of form (x^2+11*y^2)/4.
Also, primes of the form x^2-xy+3y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes congruent to 0, 1, 3, 4, 5 or 9 (mod 11). As this discriminant has class number 1, all binary quadratic forms ax^2+bxy+cy^2 with b^2-4ac=-11 represent these primes. - Rick L. Shepherd, Jul 25 2014
Also, primes which are squares (mod 11) (or, (mod 22), cf. A191020). - M. F. Hasler, Jan 15 2016
Also, primes p such that Legendre(-11,p) = 0 or 1. - N. J. A. Sloane, Dec 25 2017

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 4000 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002346 and A002347 for values of x and y.

Primes in A028954.

QuadPrimes2[1, 1, 3, 100000] (* see A106856 *)

{ fc2(a,b,c,M) = my(p,t1,t2,n);
m = 0;
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, m++; print(m," ",p )));
}
fc2(1,-1,3,10703);

Edited by N. J. A. Sloane, Jun 01 2014 and Jun 16 2014

A106865 Primes of the form 2x^2 + 2xy + 3y^2.

Original entry on oeis.org

2, 3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -20.
Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
Primes congruent to 2, 3, 7 modulo 20. - Michael Somos, Aug 13 2006
In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - Alonso del Arte, Dec 19 2015

			x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

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)

For n > 1, a(n) = A122870(n-1). Cf. A122870, A106864.

select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # Robert Israel, Dec 23 2015

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

is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ Charles R Greathouse IV, Feb 09 2017

Complement(A000040, A020669).

A033201 Primes of the form x^2 + 10*y^2.

Original entry on oeis.org

11, 19, 41, 59, 89, 131, 139, 179, 211, 241, 251, 281, 331, 379, 401, 409, 419, 449, 491, 499, 521, 569, 571, 601, 619, 641, 659, 691, 739, 761, 769, 809, 811, 859, 881, 929, 971, 1009, 1019, 1049, 1051, 1091, 1129, 1171, 1201, 1249, 1259, 1289, 1291, 1321, 1361, 1409, 1451, 1459, 1481

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 36.

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)

Cf. A139643.

Primes in A020673.

[p: p in PrimesUpTo(1500) | NormEquation(10,p) eq true]; // Bruno Berselli, Jul 03 2016

Clear[f,lst,p,x,y]; f[x_,y_]:=x^2+10*y^2; lst={};Do[Do[p=f[x,y];If[PrimeQ[p]&&p<7212,AppendTo[lst,p]],{y,0,6!}],{x,0,6!}];Take[Union[lst],222] (* Vladimir Joseph Stephan Orlovsky, Aug 04 2009 *)
QuadPrimes2[1, 0, 10, 10000] (* see A106856 *)

select(n->vecsearch([1,9,11,19],n%40), primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

Same as primes congruent to 1, 9, 11, or 19 mod 40. See, e.g., Cox, p. 36.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

A106867 Primes of the form 2x^2 + xy + 3*y^2.

Original entry on oeis.org

2, 3, 13, 29, 31, 41, 47, 71, 73, 127, 131, 139, 151, 163, 179, 193, 197, 233, 239, 257, 269, 277, 311, 331, 349, 353, 397, 409, 439, 443, 461, 487, 491, 499, 509, 541, 547, 577, 587, 601, 647, 653, 673, 683, 739, 761, 811, 823, 857, 859, 863, 887, 929, 947

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -23.
Primes p such that the polynomial x^3-x-1 is irreducible over Zp. The polynomial discriminant is also -23. - T. D. Noe, May 13 2005
Also, primes p such that tau(p) = A000594(p) == -1 (mod 23). [A proof can probably be found in van der Blij (1952). Thanks to Juan Arias-de-Reyna for this reference. - N. J. A. Sloane, Nov 29 2016]

F. van der Blij, Binary quadratic forms of discriminant -23. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 498-503; Math. Rev. MR0052462.
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.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. R. Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)).
Cf. also A000594.
These are the primes in A028929.

Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *)

forprime(p=2,10^4,if(0==#polrootsmod(x^3-x-1,p),print1(p,", "))); /* Joerg Arndt, Jul 27 2011 */

forprime(p=2,10^4,if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* Joerg Arndt, Mar 30 2013 */

from itertools import count, islice
from sympy import prime, GF, Poly
from sympy.abc import x
def A106867_gen(): # generator of terms
    return filter(lambda p:Poly(x**3-x-1,domain=GF(p)).is_irreducible, (prime(i) for i in count(1)))
A106867_list = list(islice(A106867_gen(),20)) # Chai Wah Wu, Nov 11 2022

A107003 Primes of the form 24n + 5.

Original entry on oeis.org

5, 29, 53, 101, 149, 173, 197, 269, 293, 317, 389, 461, 509, 557, 653, 677, 701, 773, 797, 821, 941, 1013, 1061, 1109, 1181, 1229, 1277, 1301, 1373, 1493, 1613, 1637, 1709, 1733, 1877, 1901, 1949, 1973, 1997, 2069, 2141, 2213, 2237, 2309, 2333, 2357, 2381, 2477

Offset: 1

T. D. Noe, May 09 2005

Primes of the form 5x^2+2xy+5y^2, with x and y any integer. Discriminant=-96. Also primes of the forms 5x^2+4xy+20y^2 and 5x^2+2xy+29y^2. See A140633. - T. D. Noe, May 19 2008

Also primes of the form -4*x^2+4*x*y+5*y^2, of discriminant -96 (as well as of the form 8*x^2+16*x*y+5*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 28 2008

			29 is a member because we can write 29=-4*4^2+4*4*3+5*3^2 (or 29=8*1^2+16*1*1+5*1^2).

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

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. A141373, A141375, A141376 (d = -96).

Union[QuadPrimes2[5, 2, 5, 10000], QuadPrimes2[5, -2, 5, 10000]] (* see A106856 *)
Select[24*Range[0,200]+5,PrimeQ] (* Harvey P. Dale, Aug 25 2025 *)

select(n->n%24==5, primes(1000)) \\ Charles R Greathouse IV, Dec 07 2014

a(n) ~ 8n log n. - Charles R Greathouse IV, Dec 07 2014

Name and comment switched by Charles R Greathouse IV, Dec 07 2014

Edited by N. J. A. Sloane, Jul 14 2019

cp's OEIS Frontend

A033215 Primes of form x^2+21*y^2.

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

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A107006 Primes of the form 4x^2-4xy+7y^2, with x and y nonnegative.

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A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

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Magma

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A033210 Primes of the form x^2+13*y^2.

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A056874 Primes of form x^2+xy+3y^2, discriminant -11.

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A106865 Primes of the form 2x^2 + 2xy + 3y^2.

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Maple

A106867 Primes of the form 2x^2 + xy + 3*y^2.