A040098 -id:A040098 - cp's OEIS frontend

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1151

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Primes that are the sum of no fewer than four positive squares.
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Primes p such that x^4 = 2 has just two solutions mod p. Subsequence of A040098. Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 = 2 if and only if p - i is a solution mod p of x^4 = 2, so the sum of the two solutions is p. The solutions are given in A065907 and A065908. - Klaus Brockhaus, Nov 28 2001
As this is a subset of A001132, this is also a subset of the primes of form x^2 - 2y^2. And as this is also a subset of A038873, this is also a subset of the primes of form x^2 - 2y^2. - Tito Piezas III, Dec 28 2008
Subsequence of A141164. - Reinhard Zumkeller, Mar 26 2011
Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - Arkadiusz Wesolowski, Apr 05 2012
Primes p such that p XOR 6 = p - 6. - Brad Clardy, Jul 22 2012

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A004771.
Cf. A040098, A014754, A065907, A065908, A010051.
Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).

a007522 n = a007522_list !! (n-1)
a007522_list = filter ((== 1) . a010051) a004771_list
-- Reinhard Zumkeller, Jan 29 2013

[p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // Vincenzo Librandi, Jun 26 2014

select(isprime, [seq(i,i=7..10000,8)]); # Robert Israel, Nov 22 2016

Select[8Range[200] - 1, PrimeQ] (* Alonso del Arte, Nov 07 2016 *)

(A007522(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z == 2, print1(p, ", ")))); A007522(1400)  \\ Does not return a(m) but prints all terms <= m. - Edited to make it executable by M. F. Hasler, May 22 2025.

A007522_upto(N, start=1)=select(p->p%8==7, primes([start, N]))
#A7522=A007522_upto(10^5)
A007522(n)={while(#A7522A007522_upto(N*3\2, N+1))); A7522[n]} \\ M. F. Hasler, May 22 2025

Equals A000040 INTERSECT A004215. - R. J. Mathar, Nov 22 2006

a(n) = 7 + A139487(n)*8, n >= 1. - Wolfdieter Lang, Feb 18 2015

A038873 Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8.

Original entry on oeis.org

2, 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599, 601, 607, 617

Offset: 1

N. J. A. Sloane

Same as A001132 except for initial term.
Primes p such that x^2 = 2 has a solution mod p.
The primes of the form x^2 + 2xy - y^2 coincide with this sequence. These are also primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008
Therefore these are composite in Z[sqrt(2)], as they can be factored as (u^2 - 2v^2)*(u^2 + 2v^2). - Alonso del Arte, Oct 03 2012
After a(1) = 2, these are the primes p such that p^4 == 1 (mod 96). - Gary Detlefs, Jan 22 2014
Also primes of the form 2v^2 - u^2. For example, 23 = 2*4^2 - 3^2. - Jerzy R Borysowicz, Oct 27 2015
Prime factors of A008865 and A028884. - Klaus Purath, Dec 07 2020

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Theorem 5-5, p. 68.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
K. S. Brown, Pythagorean graphs.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index entries for related sequences

Cf. A057126, A087780, A226523, A003629 (complement).
Primes in A035251.
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(617) | IsSquare(R!2) where R:=ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

seq(`if`(member(ithprime(n) mod 8, {1,2,7}),ithprime(n),NULL),n=1..113); # Nathaniel Johnston, Jun 26 2011

fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; Select[ Prime[Range[114]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)

is(n)=isprime(n) && issquare(Mod(2,n)) \\ Charles R Greathouse IV, Apr 23 2015

is(n)=abs(centerlift(Mod(n,8)))<3 && isprime(n) \\ Charles R Greathouse IV, Nov 14 2017

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

A001132 Primes == +-1 (mod 8).

Original entry on oeis.org

7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599

Offset: 1

N. J. A. Sloane

Primes p such that 2 is a quadratic residue mod p.
Also primes p such that p divides 2^((p-1)/2) - 1. - Cino Hilliard, Sep 04 2004
A001132 is exactly formed by the prime numbers of A118905: in fact at first every prime p of A118905 is p = u^2 - v^2 + 2uv, with for example u odd and v even so that p - 1 = 4u'(u' + 1) + 4v'(2u' + 1 - v') when u = 2u' + 1 and v = 2v'. u'(u' + 1) is even and v'(2u' + 1 - v') is always even. At second hand if p = 8k +- 1, p has the shape x^2 - 2y^2; letting u = x - y and v = y, comes p = (x - y)^2 - y^2 + 2(x - y)y = u^2 - v^2 + 2uv so p is a sum of the two legs of a Pythagorean triangle. - Richard Choulet, Dec 16 2008
These are also the primes of form x^2 - 2y^2, excluding 2. See A038873. - Tito Piezas III, Dec 28 2008
Primes p such that p^2 mod 48 = 1. - Gary Detlefs, Dec 29 2011
Primes in A047522. - Reinhard Zumkeller, Jan 07 2012
This sequence gives the odd primes p which satisfy C(p, x = 0) = +1, where C(p, x) is the minimal polynomial of 2*cos(Pi/p) (see A187360). For the proof see a comment on C(n, 0) in A230075.  - Wolfdieter Lang, Oct 24 2013
Each a(n) corresponds to precisely one primitive Pythagorean triangle. For a proof see the W. Lang link, also for a table. See also the comment by Richard Choulet above, where the case u even and v odd has not been considered. - Wolfdieter Lang, Feb 17 2015
Primes p such that p^2 mod 16 = 1. - Vincenzo Librandi, May 23 2016
Rational primes that decompose in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (2/p)
Wolfdieter Lang, A001132: Primes +1 (mod 8) or -1 (mod 8) and Sum of Legs of Primitive Pythagorean Triangles.
Index entries for related sequences
Index to sequences related to decomposition of primes in quadratic fields

For primes p such that x^m = 2 (mod p) has a solution see A001132 (for m = 2), A040028 (m = 3), A040098 (m = 4), A040159 (m = 5), A040992 (m = 6), A042966 (m = 7), A045315 (m = 8), A049596 (m = 9), A049542 (m = 10) - A049595 (m = 63). Jeff Lagarias (lagarias(AT)umich.edu) points out that all these sequences are different, although this may not be apparent from looking just at the initial terms.
Agrees with A038873 except for initial term.
Cf. A118905, A010051.
Union of A007519 and A007522.

a001132 n = a001132_list !! (n-1)
a001132_list = [x | x <- a047522_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[p: p in PrimesUpTo (600) | p^2 mod 16 eq 1]; // Vincenzo Librandi, May 23 2016

seq(`if`(member(ithprime(n) mod 8, {1,7}),ithprime(n),NULL),n=1..109); # Nathaniel Johnston, Jun 26 2011
for n from 1 to 600 do if (ithprime(n)^2 mod 48 = 1) then print(ithprime(n)) fi od. # Gary Detlefs, Dec 29 2011

Select[Prime[Range[250]], MemberQ[{1, 7}, Mod[#, 8]] &]  (* Harvey P. Dale, Apr 29 2011 *)
Select[Union[8Range[100] - 1, 8Range[100] + 1], PrimeQ] (* Alonso del Arte, May 22 2016 *)

select(p->p%8==1 ||p%8==7, primes(100)) \\ Charles R Greathouse IV, May 18 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, May 18 2015

A040028 Primes p such that x^3 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 433

Offset: 1

N. J. A. Sloane

This is the union of {3}, A003627 (primes congruent to 2 mod 3) and A014752 (primes of the form x^2+27y^2). By Thm. 4.15 of [Cox], p is of the form x^2+27y^2 if and only if p is congruent to 1 mod 3 and 2 is a cubic residue mod p. If p is not congruent to 1 mod 3, then every number is a cubic residue mod p, including 2. - Andrew V. Sutherland, Apr 26 2008

Complement of A040034 relative to A000040. - Vincenzo Librandi, Sep 13 2012

David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.
Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory", second ed., 1990, Springer-Verlag.

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

Cf. A001132. Number of primes p < 10^n for which 2 is a cubic residue (mod p) is in A097142.
Cf. A000040, A003627, A014752, A040034.
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(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; // Klaus Brockhaus, Dec 02 2008

f[p_] := Block[{k = 2}, While[k < p && Mod[k^3, p] != 2, k++ ]; If[k == p, 0, 1]]; Select[ Prime[ Range[100]], f[ # ] == 1 &] (* Robert G. Wilson v, Jul 26 2004 *)

select(p->ispower(Mod(2,p),3),primes(100)) \\ Charles R Greathouse IV, Apr 28 2015

a(n) ~ (3/2) n log n. - Charles R Greathouse IV, Apr 06 2022

Typo corrected to A014752 by Paul Landon (paullandon(AT)hotmail.com), Jan 25 2010

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

Original entry on oeis.org

7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039, 1063

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Or, primes p which do not divide 2^n+1 for any n.
The possibility n=0 in the above rules out A072936(1)=2; apart from this, a(n)=A072936(n+1). - M. F. Hasler, Dec 08 2007
The order of 2 mod p is odd iff 2^k=1 mod p, where p-1=2^s*k, k odd. - M. F. Hasler, Dec 08 2007
Has density 7/24 (Hasse).
From Jianing Song, Jun 27 2025: (Start)
The multiplicative order of 2 modulo a(n) is A139686(n).
Contained in primes congruent to 1 or 7 modulo 8 (primes p such that 2 is a quadratic residue modulo p, A001132), and contains primes congruent to 7 modulo 8 (A007522). (End)

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.
P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
H. H. Hasse, Über die Dichte der Primzahlen p, ... , Math. Ann., 168 (1966), 19-23.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
Chunlei Li, Nian Li, and Matthew G. Parker, Complementary Sequence Pairs of Types II and III. [From _N. J. A. Sloane_, Jun 16 2012]

Cf. Complement in primes of A091317.
Cf. A001132, A007522, A040098, A045315, A049564, A139686 (the actual multiplicative orders).
Cf. Essentially the same as A072936 (except for missing leading term 2).
Cf. other bases: this sequence (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

okQ[p_] := OddQ[MultiplicativeOrder[2, p]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)

isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])
listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ M. F. Hasler, Dec 08 2007

lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")););} \\ Michel Marcus, Feb 06 2015

Edited by M. F. Hasler, Dec 08 2007

More terms from Max Alekseyev, Feb 06 2010

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

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 *)

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 *)

A014754 Primes p == 1 mod 8 such that 2 and -2 are both 4th powers (one implies other) mod p.

Original entry on oeis.org

73, 89, 113, 233, 257, 281, 337, 353, 577, 593, 601, 617, 881, 937, 1033, 1049, 1097, 1153, 1193, 1201, 1217, 1249, 1289, 1433, 1481, 1553, 1601, 1609, 1721, 1753, 1777, 1801, 1889, 1913, 2089, 2113, 2129, 2273, 2281, 2393, 2441, 2473, 2593, 2657, 2689

Offset: 1

Warren D. Smith

nonn

Primes p such that x^4 == 2 has more than two (in fact four) solutions mod p. This is the sequence of terms common to A040098 (primes p such that x^4 == 2 has a solution mod p) and A007519 (primes of form 8n+1). Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 == 2 iff p - i is a solution mod p of x^4 == 2, thus the sum of first and fourth solution is p and so is the sum of second and third solution. The solutions are given in A065909, A065910, A065911 and A065912. - Klaus Brockhaus, Nov 28 2001

Primes of the form x^2+64y^2. - T. D. Noe, May 13 2005

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..9769 (the first 1000 terms were found by Vincenzo Librandi)

Cf. A040098, A007519, A014754, A007522, A065909, A065910, A065911, A065912, A070179.

A014754(m) = local(p,s,x,z); forprime(p = 3,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); z = matsize(s)[2]; if(z>2,print1(p,",")))

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

forprime(p=1, 9999, p%8==1&&ispower(Mod(2, p), 4)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

is_A014754(p)={p%8==1&&ispower(Mod(2, p), 4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Removed erroneous Mma program; extended b-file using first PARI program of M. F. Hasler. - N. J. A. Sloane, Jun 06 2014

cp's OEIS Frontend

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

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A038873 Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8.

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A001132 Primes == +-1 (mod 8).

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A040028 Primes p such that x^3 = 2 has a solution mod p.

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A014663 Primes p such that multiplicative order of 2 modulo p is odd.

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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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