A040101 -id:A040101 - cp's OEIS frontend

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

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

A385220 Primes p such that multiplicative order of 3 modulo p is odd.

Original entry on oeis.org

2, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709, 719, 733, 743, 757, 827, 829, 839, 863, 887, 911, 947, 971, 983

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of 3 modulo a(n) is A385226(n).
Without 2, contained in primes congruent to 1 or 11 modulo 12 (primes p such that 3 is a quadratic residue modulo p; A097933), and contains primes congruent to 11 modulo 12 (A068231).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

A068231 < this sequence < A045317 < A040101 < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.
Complement of A301916 in {primes} \ {3}.
Cf. A385226 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), this sequence (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[3, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385220(p) = isprime(p) && (p!=3) && znorder(Mod(3,p))%2

A045317 Primes p such that x^8 = 3 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709

Offset: 1

N. J. A. Sloane

Complement of A045318 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Union of 2, 5, A068231 (primes congruent to 11 modulo 12), prime p == 5 (mod 8) such that 3^((p-1)/4) == 1 (mod p), and primes p == 1 (mod 8) such that 3^((p-1)/8) == 1 (mod p). - Jianing Song, Jun 22 2025

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

Cf. A000040, A045318.

A068231 < A385220 < this sequence < A040101 < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 3}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^8- 3, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

isok(p) = isprime(p) && ispower(Mod(3, p), 8); \\ Michel Marcus, Oct 17 2018

isA045317(p) = isprime(p) && (p==2 || p==3 || p%12==11 || (p%8==5 && Mod(3, p)^((p-1)/4) == 1) || (p%8==1 && Mod(3, p)^((p-1)/8) == 1)) \\ Jianing Song, Jun 22 2025

A040102 Primes p such that x^4 = 3 has no solution mod p.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367

Offset: 1

N. J. A. Sloane

Complement of A040101 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^4 eq 3} ]; // Vincenzo Librandi, Sep 17 2012

ok[p_]:= Reduce[Mod[x^4 - 3, p] == 0, x, Integers] == False;Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 17 2012  *)

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

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A385220 Primes p such that multiplicative order of 3 modulo p is odd.

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A045317 Primes p such that x^8 = 3 has a solution mod p.

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A040102 Primes p such that x^4 = 3 has no solution mod p.

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