A038873 -id:A038873 - cp's OEIS frontend

A040098 Primes p such that x^4 = 2 has a solution mod p.

2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 113, 127, 151, 167, 191, 199, 223, 233, 239, 257, 263, 271, 281, 311, 337, 353, 359, 367, 383, 431, 439, 463, 479, 487, 503, 577, 593, 599, 601, 607, 617, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919

Offset: 1

N. J. A. Sloane

For a prime p congruent to 1 mod 8, 2 is a biquadratic residue mod p if and only if there are integers x,y such that x^2 + 64*y^2 = p. 2 is also a biquadratic residue mod 2 and mod p for any prime p congruent to 7 mod 8 and for no other primes. - Fred W. Helenius (fredh(AT)ix.netcom.com), Dec 30 2004

Complement of A040100 relative to A000040. - Vincenzo Librandi, Sep 13 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
Franz Lemmermeyer, Bibliography on Reciprocity Laws
Index entries for related sequences

Cf. A000040, A001132, A040028, A040100, A045315.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[ p: p in PrimesUpTo(919) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 2} ]; // Klaus Brockhaus, Dec 02 2008

ok[p_] := Reduce[ Mod[x^4 - 2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200]], ok] (* Jean-François Alcover, Dec 14 2011 *)

forprime(p=2,2000,if([]~!=polrootsmod(x^4-2,p),print1(p,", ")));print(); \\ Joerg Arndt, Jul 27 2011

A028884 a(n) = (n + 3)^2 - 8.

Original entry on oeis.org

1, 8, 17, 28, 41, 56, 73, 92, 113, 136, 161, 188, 217, 248, 281, 316, 353, 392, 433, 476, 521, 568, 617, 668, 721, 776, 833, 892, 953, 1016, 1081, 1148, 1217, 1288, 1361, 1436, 1513, 1592, 1673, 1756, 1841, 1928, 2017, 2108, 2201, 2296, 2393

Offset: 0

Patrick De Geest

From Klaus Purath, Jan 04 2023: (Start)
The product of two consecutive terms belongs to the sequence: a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-n-1) + 1.
a(n) is never divisible by primes given in A003629.
Each odd prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -6 (mod p).
The prime factors are listed in A038873 and the primes in A028886.
For n > 0, this is a proper subsequence of A079896.
Conjecture: a(n) = A079896(A265284(n-1)). -
(End)

			From _Stefano Spezia_, Nov 08 2022: (Start)
Illustrations for n = 0..4:
          *       * * *     * * * * *
      a(0) = 1    *   *     *       *
                  * * *     *   *   *
                a(1) = 8    *       *
                            * * * * *
                            a(2) = 17
.
   * * * * * * *    * * * * * * * * *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   * * * * * * *    *   *   *   *   *
     a(3) = 28      *               *
                    * * * * * * * * *
                        a(4) = 41
(End)

Altug Alkan, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005563, A014616, A028560, A082111, A190576.

a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013

Range[3, 50]^2 - 8 (* Alonso del Arte, Aug 15 2016 *)

a(n)=(n+3)^2-8 \\ Charles R Greathouse IV, Oct 07 2015

(3 to 49).map(n => n * n - 8) // Alonso del Arte, May 07 2020

a(n) = a(n-1) + 2*n + 5 (with a(0) = 1). - Vincenzo Librandi, Aug 05 2010
a(n) = A028560(n) + 1; A014616(n) = floor(a(n+1)/4). - Reinhard Zumkeller, Apr 07 2013
G.f.: (-1 - 5*x + 4*x^2)/(x - 1)^3. - R. J. Mathar, Mar 24 2013
Sum_{n >= 0} 1/a(n) = 51/112 - Pi*cot(2*Pi*sqrt(2))/(4*sqrt(2)) = 1.3839174974448... . - Vaclav Kotesovec, Apr 10 2016
E.g.f.: (1 + 7*x + x^2)*exp(x). - G. C. Greubel, Aug 19 2017
Sum_{n >= 0} (-1)^n/a(n) = (-19 + 14*sqrt(2)*Pi*cosec(2*sqrt(2)*Pi))/112. - Amiram Eldar, Nov 04 2020
From Klaus Purath, Jan 04 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2, n >= 2.
a(n) = A082111(n) + n.
a(n) = A190576(n+1) - n. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 7*Pi/(45*sqrt(2)*sin(2*sqrt(2)*Pi)).
Product_{n>=0} (1 + 1/a(n)) = (4*sqrt(14)/9)*sin(sqrt(7)*Pi)/sin(2*sqrt(2)*Pi). (End)

Definition corrected by Omar E. Pol, Jul 27 2009

A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p.

Original entry on oeis.org

17, 41, 97, 137, 193, 241, 313, 401, 409, 433, 449, 457, 521, 569, 641, 673, 761, 769, 809, 857, 929, 953, 977, 1009, 1129, 1297, 1321, 1361, 1409, 1489, 1657, 1697, 1873, 1993, 2017, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713

Offset: 1

Klaus Brockhaus, Apr 29 2002

Complement of A014754 with regard to primes of the form 8*k+1.
These appear to be the primes p for which 4^((p-1)*n/8) mod p = (p-2)*( n mod 2)+1. For example, 4^(5*n) mod 41 = 1,40,1,40,1,40...= 39*(n mod 2)+1 and 4^(30*n) mod 241 = 1,240,1,240,1,240...= 239*(n mod 2) +1. - Gary Detlefs, Jul 06 2014
Primes p == 1 mod 8 such that 2^((p-1)/4) == -1 mod p. - Robert Israel, Jul 06 2014
A very similar sequence is A293394. - Jonas Kaiser, Nov 08 2017

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A038873, A040098, A040100, A059667, A070180-A070188, A014754.

[p: p in PrimesUpTo(3000) | not exists{x: x in ResidueClassRing(p) | x^4 eq 2} and exists{x: x in ResidueClassRing(p) | x^2 eq 2}]; // Vincenzo Librandi, Sep 21 2012

select(p -> isprime(p) and 2 &^((p-1)/4) mod p = p-1, [8*k+1$k=1..10000]); # Robert Israel, Jul 06 2014

PARI
```
forprime(p=2,2720,x=0; while(x
				
```

{a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( cMichael Somos, Mar 22 2008 */

ok(p, r, k1, k2)={
if ( Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
if ( Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
return(1);
}
forprime(p=2,10^4, if (ok(p,2,2,2^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

is(n)=n%8==1 && Mod(2,n)^(n\4)==-1 && isprime(n) \\ Charles R Greathouse IV, Nov 10 2017

Primes of the form 8*k + 1 but not x^2 + 64*y^2. - Michael Somos, Mar 22 2008

a(n) ~ 8n log n. - Charles R Greathouse IV, Nov 10 2017

A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25, 28, 31, 32, 34, 36, 41, 46, 47, 49, 50, 56, 62, 63, 64, 68, 71, 72, 73, 79, 81, 82, 89, 92, 94, 97, 98, 100, 103, 112, 113, 119, 121, 124, 126, 127, 128, 136, 137, 142, 144, 146, 151, 153, 158, 161, 162, 164, 167, 169, 175, 178

Offset: 1

N. J. A. Sloane

nonn

x^2 - 2y^2 has discriminant 8. - N. J. A. Sloane, May 30 2014
A positive number n is representable in the form x^2 - 2y^2 iff every prime p == 3 or 5 (mod 8) dividing n occurs to an even power.
Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m=2 (A035185). [amended by Georg Fischer, Sep 03 2020]
Also positive numbers of the form 2x^2 - y^2. If x^2 - 2y^2 = n, 2(x+y)^2 - (x+2y)^2 = n. - Franklin T. Adams-Watters, Nov 09 2009
Except 2, prime numbers in this sequence have the form p=8k+-1. According to the first comment, prime factors of the forms (8k+-3),(8k+-5) occur in x^2 - 2y^2 in even powers. If x^2 - 2y^2 is a prime number, those powers must be 0. Only factors 8k+-1 remain. Example: 137=8*17+1. - Jerzy R Borysowicz, Nov 04 2015
The product of any two terms of the sequence is a term too. A proof follows from the identity: (a^2-2b^2)(c^2-2d^2) = (2bd+ac)^2 - 2(ad+bc)^2. Example: 127*175 has form x^2-2y^2, with x=9335, y=6600. - Jerzy R Borysowicz, Nov 28 2015
Primitive terms (not a product of earlier terms that are greater than 1 in the sequence) are A055673 except 1. - Charles R Greathouse IV, Sep 10 2016
Positive numbers of the form u^2 + 2uv - v^2. - Thomas Ordowski, Feb 17 2017
For integer numbers z, a, k and z^2+a^2>0, k>=0: z^(4k) + a^4 is in A035251 because z^(4k) + a^4 = (z^(2k) + a^2)^2 - 2(a*z^k)^2. Assume 0^0 = 1. Examples: 3^4 + 1^4 = 82, 3^8+4^4=6817. - Jerzy R Borysowicz, Mar 09 2017
Numbers that are the difference between two legs of a Pythagorean right triangle. - Michael Somos, Apr 02 2017

			The (x,y) pairs, with minimum x, that solve the equation are (1,0), (2,1), (2,0), (3,1), (4,2), (3,0), (4,1), (4,0), (5,2), (6,3), (5,1), (5,0), (6,2), (7,3), (8,4), (6,1), (6,0), (7,2), (8,3), (7,1), (7,0), (10,5), (8,2), ... If the positive number is a perfect square, y=0 yields a trivial solution. - _R. J. Mathar_, Sep 10 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Matthews, Thue's theorem and the diophantine equation x^2-D*y^2=+-N, Math. Comp. 71 (239) (2002) 1281-1286.
K. Matthews, Solving the diophantine equation x^2-D*y^2=N, D>0, (2016).
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 [Edited and cached copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A035185, A042965, A001481, A000047.
Primes: A038873.
Complement of A232531. - Thomas Ordowski and Altug Alkan, Feb 09 2017

filter:= proc(n) local F;
  F:= select(t -> t[1] mod 8 = 3 or t[1] mod 8 = 5, ifactors(n)[2]);
  map(t -> t[2],F)::list(even);
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 01 2015

Reap[For[n = 1, n < 200, n++, r = Reduce[x^2 - 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

select(x -> x, direuler(p=2,201,1/(1-(kronecker(2,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 17 2006 */

is(n)=#bnfisintnorm(bnfinit(z^2-2),n) \\ Ralf Stephan, Oct 14 2013

from itertools import count, islice
from sympy import factorint
def A035251_gen(): # generator of terms
    return filter(lambda n:all(not((2 < p & 7 < 7) and e & 1) for p, e in factorint(n).items()),count(1))
A035251_list = list(islice(A035251_gen(),30)) # Chai Wah Wu, Jun 28 2022

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

A045331 Primes congruent to {1, 2, 3} mod 6; or, -3 is a square mod p.

Original entry on oeis.org

2, 3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane

-3 is a quadratic residue mod a prime p iff p is in this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Apart from initial term, same as A007645; apart from initial two terms, same as A002476.
Cf. A002313, A033203, A038873, A038874, A057128.
Subsequence of A047246.

a045331 n = a045331_list !! (n-1)
a045331_list = filter ((< 4) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

[p: p in PrimesUpTo(700) | p mod 6 in [1, 2, 3]]; // Vincenzo Librandi, Aug 08 2012

Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,6]]&]  (* Harvey P. Dale, Mar 31 2011 *)
Join[{2,3},Select[Range[7,10^3,6],PrimeQ]] (* Zak Seidov, May 20 2011 *)

select(n->n%6<5,primes(100)) \\ Charles R Greathouse IV, May 20 2011

More terms from Henry Bottomley, Aug 10 2000

A040159 Primes p such that x^5 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 151, 157, 163, 167, 173, 179, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 283, 293, 307, 313, 317, 337, 347, 349, 353

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n=1..1000
Index entries for related sequences

Cf. A001132, A040028, A040098, A040160.
Has same beginning as A042991 but is strictly different.
For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[p: p in PrimesUpTo(400) | exists{x: x in ResidueClassRing(p) | x^5 eq 2}]; // Bruno Berselli, Sep 12 2012

ok [p_]:=Reduce[Mod[x^5- 2, p]== 0, x, Integers]=!= False; Select[Prime[Range[180]], ok] (* Vincenzo Librandi, Sep 12 2012 *)

A025021 Numbers whose least quadratic nonresidue (A020649) is 3.

Original entry on oeis.org

7, 14, 17, 31, 34, 41, 49, 62, 79, 82, 89, 98, 103, 113, 119, 127, 137, 151, 158, 161, 178, 199, 206, 217, 223, 226, 233, 238, 254, 257, 271, 274, 281, 287, 289, 302, 322, 329, 343, 353, 367, 391, 398, 401, 434, 439, 446, 449, 463, 466, 487, 497, 511, 514, 521, 527, 542

Offset: 1

David W. Wilson

nonn

n such that n is not divisible by 4, all primes dividing n are in A038873, and at least one prime dividing n is in A003630. - Robert Israel, Jul 19 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quadratic Nonresidue.

Cf. A020649, A025020, A025022, A025023, A025024, A025025, A025026, A025027, A025028, A025029.

Cf. A003630, A038873.

select(t -> numtheory:-quadres(2,t) = 1 and numtheory:-quadres(3,t)=-1, [$1..1000]); # Robert Israel, Jul 19 2017

Select[Range[500], Min @ Complement[Range[# - 1], Mod[Range[#/2]^2, #]] == 3 &] (* Amiram Eldar, Oct 31 2020 *)

residue(n,m)={local(r);r=0;for(i=1,floor(m/2),if(i^2%m==n,r=1));r}
isA025021(n)=residue(2,n) && !residue(3,n) \\ Michael B. Porter, Apr 18 2010

A038874 Primes p such that 3 is a square mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601

Offset: 1

N. J. A. Sloane

Also primes congruent to {1, 2, 3, 11} mod 12.
The subsequence p = 1 (mod 4) corresponds to A068228 and only these entries of a(n) are squares mod 3 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Largest prime factors of n^2 - 3. - Vladimir Joseph Stephan Orlovsky, Aug 12 2009
Aside from 2 and 3, primes p such that Legendre(3, p) = 1. Bolker asserts there are infinitely many of these primes. - Alonso del Arte, Nov 25 2015
The associated bases of the squares are 1, 0, 5, 4, 7, 15, 12, 11, 8, 28, 21, 13...: 1^2 = 3 -1*2, 0^2 = 3-1*3, 5^2 = 3+ 2*11, 4^2 = 3+1*13, 7^2 = 3+2*23, 15^2 = 3+6*37, 12^2 = 3+3*47,... - R. J. Mathar, Feb 23 2017

			11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.

Cf. A002313, A033203, A038873, A045331, A057125.

If the first two terms are omitted we get A097933. A040101 is another sequence.

[p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012

select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # Robert Israel, Dec 23 2015

Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* Vincenzo Librandi, Aug 08 2012 *)
Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* Alonso del Arte, Nov 25 2015 *)

forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

More terms from Henry Bottomley, Aug 10 2000

A040992 Primes p such that x^6 = 2 has a solution mod p.

Original entry on oeis.org

2, 17, 23, 31, 41, 47, 71, 89, 113, 127, 137, 167, 191, 223, 233, 239, 257, 263, 281, 311, 353, 359, 383, 401, 431, 433, 439, 449, 457, 479, 503, 521, 569, 593, 599, 601, 617, 641, 647, 719, 727, 743, 761, 809, 839, 857, 863, 881, 887, 911, 919, 929, 953

Offset: 1

N. J. A. Sloane

Complement of A040993 relative to A000040. - Vincenzo Librandi, Sep 13 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A040993.

For primes p such that x^m == 2 (mod p) has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[p: p in PrimesUpTo(1000) | exists(t){x : x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^6- 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

forprime(p=2,2000,if([]~!=polrootsmod(x^6-2,p),print1(p,", ")));print();
/* Joerg Arndt, Jul 27 2011 */

A042966 Primes p such that x^7 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 199, 223, 227, 229, 233, 241, 251, 257, 263, 269, 271, 277, 283, 293, 307, 311, 313, 317, 331

Offset: 1

N. J. A. Sloane

Coincides with sequence of "primes p such that x^49 = 2 has a solution mod p" for first 572 terms, then diverges.
Complement of A042967 relative to A000040. - Vincenzo Librandi, Sep 13 2012
a(98) = 631 is the first such prime that is congruent to 1 (mod 7). - Georg Fischer, Jan 06 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A042967.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[p: p in PrimesUpTo(400) | exists{x: x in ResidueClassRing(p) | x^7 eq 2}]; // Bruno Berselli, Sep 12 2012

ok[p_]:= Reduce[Mod[x^7 - 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

cp's OEIS Frontend

A040098 Primes p such that x^4 = 2 has a solution mod p.

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A028884 a(n) = (n + 3)^2 - 8.

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A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p.

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A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

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A045331 Primes congruent to {1, 2, 3} mod 6; or, -3 is a square mod p.

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A040159 Primes p such that x^5 = 2 has a solution mod p.

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