A057129 -id:A057129 - cp's OEIS frontend

A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.

2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617

Offset: 1

N. J. A. Sloane

Or, primes p such that x^2 - p*y^2 represents -1.
Primes which are not Gaussian primes (meaning not congruent to 3 mod 4).
Every Fibonacci prime (with the exception of F(4) = 3) is in the sequence. If p = 2n+1 is the prime index of the Fibonacci prime, then F(2n+1) = F(n)^2 + F(n+1)^2 is the unique representation of the prime as sum of two squares. - Sven Simon, Nov 30 2003
Except for 2, primes of the form x^2 + 4y^2. See A140633. - T. D. Noe, May 19 2008
Primes p such that for all p > 2, p XOR 2 = p + 2. - Brad Clardy, Oct 25 2011
Greatest prime divisor of r^2 + 1 for some r. - Michel Lagneau, Sep 30 2012
Empirical result: a(n), as a set, compose the prime factors of the family of sequences produced by A005408(j)^2 + A005408(j+k)^2 = (2j+1)^2 + (2j+2k+1)^2, for j >= 0, and a given k >= 1 for each sequence, with the addition of the prime factors of k if not already in a(n). - Richard R. Forberg, Feb 09 2015
Primes such that when r is a primitive root then p-r is also a primitive root. - Emmanuel Vantieghem, Aug 13 2015
Primes of the form (x^2 + y^2)/2. Note that (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2 = a^2 + b^2 with x = a + b and y = a - b. More generally, primes of the form (x^2 + y^2) / A001481(n) for every fixed n > 1. - Thomas Ordowski, Jul 03 2016
Numbers n such that ((n-2)!!)^2 == -1 (mod n). - Thomas Ordowski, Jul 25 2016
Primes p such that (p-1)!! == (p-2)!! (mod p). - Thomas Ordowski, Jul 28 2016
The product of 2 different terms (x^2 + y^2)(z^2 + v^2) = (xz + yv)^2 + (xv - yz)^2 is sum of 2 squares (A000404) because (xv - yz)^2 > 0. If x were equal to yz/v then (x^2 + y^2)/(z^2 + v^2) would be equal to ((yz/v)^2 + y^2)/(z^2 + v^2) = y^2/v^2 which is not possible because (x^2 + y^2) and (z^2 + v^2) are prime numbers. For example, (2^2 + 5^2)(4^2 + 9^2) = (2*4 + 5*9)^2 + (2*9 - 5*4)^2. - Jerzy R Borysowicz, Mar 21 2017

			13 is in the sequence since it is prime and 13 = 4*3 + 1.  Also 13 = 2^2 + 3^2.  And -1 is a square (mod 13): -1 + 2*13 = 25 = 5^2.  Of course, only the first term is congruent to 2 (mod 4). - _Michael B. Porter_, Jul 04 2016

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 872.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
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).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Dario Alpern, Online program that calculates sum of squares representation
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n+1 Theorem
G. Xiao, Two squares
Index entries for Gaussian integers and primes

Apart from initial term, same as A002144. For values of x and y see A002330 and A002331.

Cf. A033203, A038873, A038874, A045331, A008784, A057129, A084163, A084165, A137351.

a002313 n = a002313_list !! (n-1)
a002313_list = filter ((`elem` [1,2]) . (`mod` 4)) a000040_list
-- Reinhard Zumkeller, Feb 04 2014

[p: p in PrimesUpTo(700) | p mod 4 in {1,2}]; // Vincenzo Librandi, Feb 18 2015

with(numtheory): for n from 1 to 300 do if ithprime(n) mod 4 = 1 or ithprime(n) mod 4 = 2 then printf(`%d,`,ithprime(n)) fi; od:
# alternative
A002313 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    elif n = 2 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a ;
            end if;
        end do:
    end if;
end proc:
seq(A002313(n),n=1..100) ; # R. J. Mathar, Feb 01 2024

Select[ Prime@ Range@ 115, Mod[#, 4] != 3 &] (* Robert G. Wilson v *)
fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] == {}; Select[ Prime@ Range@ 115, fQ] (* Robert G. Wilson v, Dec 19 2013 *)

select(p->p%4!=3, primes(1000)) \\ Charles R Greathouse IV, Feb 11 2011

a(n) ~ 2n log n. - Charles R Greathouse IV, Jul 04 2016

a(n) = A002331(n)^2 + A002330(n)^2. See crossrefs. - Wolfdieter Lang, Dec 11 2016

More terms from Henry Bottomley, Aug 10 2000

More terms from James Sellers, Aug 22 2000

A057126 Numbers k such that 2 is a square mod k.

Original entry on oeis.org

1, 2, 7, 14, 17, 23, 31, 34, 41, 46, 47, 49, 62, 71, 73, 79, 82, 89, 94, 97, 98, 103, 113, 119, 127, 137, 142, 146, 151, 158, 161, 167, 178, 191, 193, 194, 199, 206, 217, 223, 226, 233, 238, 239, 241, 254, 257, 263, 271, 274, 281, 287, 289, 302, 311, 313, 322

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Numbers that are not multiples of 4 and for which all odd prime factors are congruent to +/- 1 mod 8. - Eric M. Schmidt, Apr 20 2013
Apparently the same as the list of numbers primitively represented by the indefinite quadratic form x^2 - 2y^2 (cf. A035251). - N. J. A. Sloane, Jun 11 2014
From Wolfdieter Lang, Jul 11 2025: (Start)
Also the negative sequence lists the numbers properly represented by the indefinite quadratic form x^2 - 2*y^2  of discriminant 4*2 = 8. For the proof see the W. Lang paper linked in A385449, Lemma 18, pp. 22-23.
The connection between the proper positive fundamental solutions (X, Y) of X^2 - 2*Y^2 = -a(n), given in A385449, and the solutions (x, y) of x^2 - 2*y^2 = a(n) is (x, y) = (2*Y - X, X - Y). If y becomes nonpositive a transformation with the matrix Mat([3,4], [2,3]) will give the positive proper fundamental solution. See the example section of A385449. See also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters for this connection, and for the matrix eq. (38) p. 14 of the mentioned linked paper.
Therefore the previous statement on the representation of a(n) is true.(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Includes the primes in A038873 and these (primes congruent to {1, 2, 7} mod 8) are the prime factors of the terms in this sequence.
Cf. A008784, A057125, A057127, A057128, A057129, A057757, A035251, A038873.
Cf. A087780 (number of solutions mod n).

with(numtheory); [seq(mroot(2,2,p),p=1..300)];

ok[n_] := Reduce[ Mod[2 - k^2, n] == 0, k, Integers] =!= False; Prepend[ Select[ Range[400], ok], 1] (* Jean-François Alcover, Sep 20 2012 *)

isok(n) = issquare(Mod(2,n)); \\ Michel Marcus, Feb 19 2016

Checked by T. D. Noe, Apr 19 2007

A057125 Numbers n such that 3 is a square mod n.

Original entry on oeis.org

1, 2, 3, 6, 11, 13, 22, 23, 26, 33, 37, 39, 46, 47, 59, 61, 66, 69, 71, 73, 74, 78, 83, 94, 97, 107, 109, 111, 118, 121, 122, 131, 138, 141, 142, 143, 146, 157, 166, 167, 169, 177, 179, 181, 183, 191, 193, 194, 213, 214, 218, 219, 222, 227, 229, 239, 241, 242

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Numbers that are not multiples of 4 or 9 and for which all prime factors greater than 3 are congruent to +/- 1 mod 12. - Eric M. Schmidt, Apr 21 2013

			3^2==3 (mod 6), so 6 is a member.

T. D. Noe, Table of n, a(n) for n=1..1000

Includes the primes in A038874 and these (primes congruent to {1, 2, 3, 11} mod 12) are the prime factors of the terms in this sequence. Cf. A008784, A057126, A057127, A057128, A057129.

Cf. A057759.

[n: n in [1..300] | exists(t){x : x in ResidueClassRing(n) | x^2 eq 3}]; // Vincenzo Librandi, Feb 20 2016

# Beware: Since 2007 at least and up to Maple 16 at least, the following Maple code returns the wrong answer for n = 6:
with(numtheory): [seq(`if`(mroot(3,2,n)=FAIL,NULL,n), n=1..400)];
# second Maple program:
with(numtheory): mroot(3, 2, 6):=3:
a:= proc(n) option remember; local m;
      for m from 1+`if`(n=1, 0, a(n-1))
      while mroot(3, 2, m)=FAIL do od; m
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 24 2017

Prepend[ Select[ Range[300], Reduce[Mod[3 - k^2, #] == 0, k, Integers] =!= False &], 1]  (* Jean-François Alcover, Sep 20 2012 *)

isok(n) = issquare(Mod(3,n)); \\ Michel Marcus, Feb 19 2016

Edited by N. J. A. Sloane, Oct 25 2008 at the suggestion of R. J. Mathar.

A057127 -2 is a square mod n.

Original entry on oeis.org

1, 2, 3, 6, 9, 11, 17, 18, 19, 22, 27, 33, 34, 38, 41, 43, 51, 54, 57, 59, 66, 67, 73, 81, 82, 83, 86, 89, 97, 99, 102, 107, 113, 114, 118, 121, 123, 129, 131, 134, 137, 139, 146, 153, 162, 163, 166, 171, 177, 178, 179, 187, 193, 194, 198, 201, 209, 211, 214, 219

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Includes the primes in A033203 and these (primes congruent to {1, 2, 3} mod 8) are the prime factors of the terms in this sequence.
Numbers that are not multiples of 4 and for which all odd prime factors are congruent to {1, 3} mod 8. - Eric M. Schmidt, Apr 21 2013
Positive integers primitively represented by x^2 + 2y^2. - Ray Chandler, Jul 22 2014
The set of the divisors of numbers of the form k^2+2. - Michel Lagneau, Jun 28 2015
The number of proper solutions (x, y) with nonnegative x of the positive definite primitive quadratic form x^2 + 2*y*2 (discriminant -8) representing a(n) is 1 for n = 1 and for n >= 2 it is 2^(P_1 + P_3), where P_1 and P_3 are the number of distinct prime divisors of a(n) congruent to 1 and 3 modulo 8, respectively. See the above comments on A033203 and this binary form. - Wolfdieter Lang, Feb 25 2021

			Binary quadratic form x^2 + 2*y^2 representing a(n), with x >= 0: a(1) = 1: one solution (x, y) = (1,0); a(2) = 2: one solution (0,1); a(3) = 3: two solutions (1, pm 1), with pm = +1 or -1; a(5) = 9 = 3^2: two solutions (1, pm 2); a(12) = 33 = 3*11: 4 solutions (1, pm 4) and (5, pm 2); a(137) = 3*11*17 = 561: eight solutions (7, pm 16), (13, pm 14), (19, pm 10) and (23, pm 4). - _Wolfdieter Lang_, Feb 25 2021

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A008784, A033203, A057125, A057126, A057128, A057129.

select(n -> numtheory:-msqrt(-2,n) <> FAIL, [$1..1000]); # Robert Israel, Jun 29 2015

Select[Range[300], IntegerQ[PowerMod[-2, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 04 2019 *)

isok(n) = issquare(Mod(-2, n)); \\ Michel Marcus, Jun 28 2015

def isA057127(n):
    if n % 4 == 0: return False
    return all(p % 8 in [1, 2, 3] for p, _ in factor(n))
[n for n in range(1, 300) if isA057127(n)]
# Eric M. Schmidt, Apr 21 2013

A057128 Numbers n such that -3 is a square mod n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 12, 13, 14, 19, 21, 26, 28, 31, 37, 38, 39, 42, 43, 49, 52, 57, 61, 62, 67, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.
Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - Eric M. Schmidt, Apr 21 2013
Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016

			a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.

Cf. A117950.

select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # Robert Israel, Feb 19 2016

Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *)

isok(n) = issquare(Mod(-3,n)); \\ Michel Marcus, Feb 19 2016

def A057128(n) :
    if n%8==0 or n%9==0: return False
    for (p, m) in factor(n) :
        if p % 6 not in [1, 2, 3] : return False
        return True
# Eric M. Schmidt, Apr 21 2013

cp's OEIS Frontend

A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.

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A057126 Numbers k such that 2 is a square mod k.

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A057125 Numbers n such that 3 is a square mod n.

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A057127 -2 is a square mod n.

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Sage

A057128 Numbers n such that -3 is a square mod n.

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Author

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Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

Sage