Displaying 1-10 of 28 results found.
Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8.
+10
71
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
COMMENTS
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
REFERENCES
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Theorem 5-5, p. 68.
MAPLE
seq(`if`(member(ithprime(n) mod 8, {1, 2, 7}), ithprime(n), NULL), n=1..113); # Nathaniel Johnston, Jun 26 2011
MATHEMATICA
fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; Select[ Prime[Range[114]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)
PROG
(Magma) [ p: p in PrimesUpTo(617) | IsSquare(R!2) where R:=ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008
CROSSREFS
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, ...
Primes == +-1 (mod 8).
(Formerly M4354 N1824)
+10
39
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
COMMENTS
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
Primes p such that p^2 mod 48 = 1. - Gary Detlefs, Dec 29 2011 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 Rational primes that decompose in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017
REFERENCES
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).
LINKS
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].
MAPLE
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
MATHEMATICA
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 *)
PROG
(Haskell)
a001132 n = a001132_list !! (n-1)
a001132_list = [x | x <- a047522_list, a010051 x == 1]
(Magma) [p: p in PrimesUpTo (600) | p^2 mod 16 eq 1]; // Vincenzo Librandi, May 23 2016
CROSSREFS
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.
Primes p such that x^4 = 2 has a solution mod p.
+10
23
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
COMMENTS
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
MATHEMATICA
ok[p_] := Reduce[ Mod[x^4 - 2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200]], ok] (* Jean-François Alcover, Dec 14 2011 *)
PROG
(Magma) [ p: p in PrimesUpTo(919) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 2} ]; // Klaus Brockhaus, Dec 02 2008 (PARI) forprime(p=2, 2000, if([]~!=polrootsmod(x^4-2, p), print1(p, ", "))); print(); \\ Joerg Arndt, Jul 27 2011
CROSSREFS
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, ...
Primes of the form x^2 + 27y^2.
+10
22
31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017
COMMENTS
Primes p == 1 (mod 3) such that 2 is a cubic residue mod p.
Primes p == 1 (mod 6) such that 2 and -2 are both cubes (one implies the other) mod p. - Warren D. SmithPrimes p = 3m+1 such that 2^m == 1 (mod p). Subsequence of A016108 which also includes composites satisfying this congruence. - Alzhekeyev Ascar M, Feb 22 2012
REFERENCES
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.
MATHEMATICA
With[{nn=50}, Take[Select[Union[First[#]^2+27Last[#]^2&/@Tuples[Range[ nn], 2]], PrimeQ], nn]] (* Harvey P. Dale, Jul 28 2014 *) nn = 1398781; re = Sort[Reap[Do[Do[If[PrimeQ[p = x^2 + 27*y^2], Sow[{p, x, y}]], {x, Sqrt[nn - 27*y^2]}], {y, Sqrt[nn/27]}]][[2, 1]]]; (* For all 17753 values of a(n), x(n) and y(n). - Zak Seidov, May 20 2016 *)
PROG
(PARI)
{ fc(a, b, c, M) = my(p, t1, t2, n); t1 = listcreate();
for(n=1, M, p = prime(n);
t2 = qfbsolve(Qfb(a, b, c), p); if(t2 == 0, , listput(t1, p)));
print(t1);
}
fc(1, 0, 27, 1000);
(PARI) list(lim)=my(v=List()); forprimestep(p=31, lim, 6, if(Mod(2, p)^(p\3)==1, listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Apr 06 2022 (Magma) [p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016
EXTENSIONS
Definition provided by T. D. Noe, May 08 2005 Defective Mma program replaced with PARI program, b-file recomputed and extended by N. J. A. Sloane, Jun 06 2014
Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.
(Formerly M5283 N2299)
+10
18
43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373, 4027, 4339, 4549, 4597, 4651, 4909, 5101, 5197, 5323, 5413, 5437, 5653, 6037
REFERENCES
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
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).
MATHEMATICA
Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[MultiplicativeOrder[2, p] == (p-1)/3, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)
PROG
(Magma) [ p: p in PrimesUpTo(4597) | r eq 1 and Order(R!2) eq q where q, r is Quotrem(p, 3) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008 (PARI) forprime(p=3, 10^4, if(znorder(Mod(2, p))==(p-1)/3, print1(p, ", "))); \\ Joerg Arndt, May 17 2013
EXTENSIONS
More terms and better definition from Don Reble, Mar 11 2006
Primes congruent to {0, 2} mod 3.
+10
17
2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563
COMMENTS
Also, primes p such that the equation x^3 == y (mod p) has a unique solution x for every choice of y. - Klaus Brockhaus, Mar 02 2001; Michel Drouzy (DrouzyM(AT)noos.fr), Oct 28 2001
MATHEMATICA
Select[Prime[Range[150]], MemberQ[{0, 2}, Mod[#, 3]]&] (* Harvey P. Dale, Jun 14 2011 *)
PROG
(Magma) [ p: p in PrimesUpTo(1000) | #[ x: x in ResidueClassRing(p) | x^3 eq 2 ] eq 1 ]; // Klaus Brockhaus, Apr 11 2009
Primes p such that x^5 = 2 has a solution mod p.
+10
12
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
MATHEMATICA
ok [p_]:=Reduce[Mod[x^5- 2, p]== 0, x, Integers]=!= False; Select[Prime[Range[180]], ok] (* Vincenzo Librandi, Sep 12 2012 *)
PROG
(Magma) [p: p in PrimesUpTo(400) | exists{x: x in ResidueClassRing(p) | x^5 eq 2}]; // Bruno Berselli, Sep 12 2012
CROSSREFS
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, ...
Primes p such that x^6 = 2 has a solution mod p.
+10
11
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
MATHEMATICA
ok[p_]:= Reduce[Mod[x^6- 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)
PROG
(PARI) forprime(p=2, 2000, if([]~!=polrootsmod(x^6-2, p), print1(p, ", "))); print();
(Magma) [p: p in PrimesUpTo(1000) | exists(t){x : x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 13 2012
CROSSREFS
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, ...
Primes p such that x^7 = 2 has a solution mod p.
+10
11
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
COMMENTS
Coincides with sequence of "primes p such that x^49 = 2 has a solution mod p" for first 572 terms, then diverges.
a(98) = 631 is the first such prime that is congruent to 1 (mod 7). - Georg Fischer, Jan 06 2022
MATHEMATICA
ok[p_]:= Reduce[Mod[x^7 - 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 13 2012 *)
PROG
(Magma) [p: p in PrimesUpTo(400) | exists{x: x in ResidueClassRing(p) | x^7 eq 2}]; // Bruno Berselli, Sep 12 2012
CROSSREFS
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, ...
Primes p such that x^8 = 2 has a solution mod p.
+10
9
2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 257, 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
COMMENTS
Coincides with the sequence of "primes p such that x^16 = 2 has a solution mod p" for first 58 terms (and then diverges).
REFERENCES
A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und -2, Deutsche Math. 4 (1939), 44-52; FdM 65 - I (1939), 112.
MATHEMATICA
ok[p_] := Reduce[ Mod[x^8-2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200] ], ok] (* Jean-François Alcover, Nov 28 2011 *)
PROG
(Magma) [p: p in PrimesUpTo(1100) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 13 2012
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