A040992 -id:A040992 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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 *)

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 *)

A040993 Primes p such that x^6 = 2 has no solution mod p.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 73, 79, 83, 97, 101, 103, 107, 109, 131, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 199, 211, 227, 229, 241, 251, 269, 271, 277, 283, 293, 307, 313, 317, 331, 337, 347, 349, 367, 373, 379, 389, 397, 409

Offset: 1

N. J. A. Sloane

Complement of A040992 relative to A000040. Coincides for the first 58 terms with A212375, that is the sequence of primes p such that x^18 = 2 has no solution mod p (first divergence is at 433, cf. A059664). Also coincides for the first 58 terms with sequence of primes p such that x^54 = 2 has no solution mod p (first divergence is at 433, cf. A059665). The sequence for x^18 and the sequence for x^54 coincide for the first 379 terms (first divergence is at 3943, cf. A059666). - Klaus Brockhaus, Feb 04 2001

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

Cf. A000040, A040992, A049550, A059664, A059665, A059666, A212375.

[p: p in PrimesUpTo(500) | forall{x: x in ResidueClassRing(p) | x^6 ne 2}]; // Bruno Berselli, Sep 13 2012

Select[Prime[Range[PrimePi[500]]], ! MemberQ[PowerMod[Range[#], 6, #], Mod[2, #]] &] (* Bruno Berselli, Sep 13 2012 *)
ok[p_] := Reduce[Mod[x^6 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[80]], ok] (* Vincenzo Librandi, Sep 21 2012 *)

A212375 added in the Brockhaus comment from Bruno Berselli, Sep 13 2012

A059664 Primes p such that x^18 = 2 has no solution mod p, but x^6 = 2 has a solution mod p.

Original entry on oeis.org

433, 919, 1423, 1999, 2017, 2143, 2287, 2791, 2953, 3457, 3889, 4177, 4519, 4663, 5113, 5167, 6679, 6967, 8713, 9631, 9649, 9721, 10009, 11863, 12241, 12583, 12799, 13177, 13591, 15913, 16057, 16111, 16561, 16921, 17551, 18127, 18793, 19081

Offset: 1

Klaus Brockhaus, Feb 04 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..340

A000040, A040992, A040993, A049550, A059665.

[p: p in PrimesUpTo(20000) | not exists{x: x in ResidueClassRing(p) | x^18 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Select[Prime[Range[PrimePi[20000]]], !MemberQ[PowerMod[Range[#], 18, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 6, #], Mod[2, #]]&] (* Vincenzo Librandi, Sep 21 2013 *)

A059665 Primes p such that x^54 = 2 has no solution mod p, but x^6 = 2 has a solution mod p.

Original entry on oeis.org

433, 919, 1423, 1999, 2017, 2143, 2287, 2791, 2953, 3457, 3889, 3943, 4177, 4519, 4663, 5113, 5167, 6679, 6967, 8713, 9631, 9649, 9721, 10009, 11287, 11863, 12241, 12583, 12799, 13177, 13591, 15913, 16057, 16111, 16561, 16921, 17551, 18127

Offset: 1

Klaus Brockhaus, Feb 04 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..389

A000040, A040992, A040993, A049550, A059666, A059664.

[p: p in PrimesUpTo(20000) | not exists{x: x in ResidueClassRing(p) | x^54 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Select[Prime[Range[PrimePi[20000]]], !MemberQ[PowerMod[Range[#], 54, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 6, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 21 2013 *)

A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

Original entry on oeis.org

17, 41, 137, 401, 433, 449, 457, 521, 569, 641, 761, 809, 857, 919, 929, 953, 977, 1361, 1409, 1423, 1657, 1697, 1999, 2017, 2081, 2143, 2153, 2287, 2297, 2417, 2609, 2633, 2729, 2753, 2777, 2791, 2801, 2897, 2953, 3041, 3209, 3329, 3457, 3593, 3617

Offset: 1

Klaus Brockhaus, Apr 29 2002

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040992, A049568, A059264, A059667, A070179, A070180, A070181, A070182, A070184, A070185, A070186, A070187, A070188.

[p: p in PrimesUpTo(5000) | not exists{x: x in ResidueClassRing(p) | x^36 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012

select(p -> isprime(p) and [msolve(x^6=2,p)]<>[] and [msolve(x^36=2,p)]=[] , [seq(i,i=3..10^4,2)]); # Robert Israel, May 13 2018

PARI
```
forprime(p=2,3700,x=0; while(x
				
```

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,2,6,6^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A070183_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        if is_nthpow_residue(2,6,p) and not is_nthpow_residue(2,36,p):
            yield p
        p = nextprime(p)
A070183_list = list(islice(A070183_gen(),20)) # Chai Wah Wu, May 02 2024

cp's OEIS Frontend

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

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

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

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A040993 Primes p such that x^6 = 2 has no solution mod p.

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