A236969 Primes p such that there exist x, y where x + y, x * y and x ^ y == 1 mod p.

3, 19, 31, 61, 67, 79, 97, 127, 307, 331, 367, 409, 457, 499, 571, 613, 631, 691, 709, 727, 733, 787, 829, 883, 991, 1063, 1087, 1093, 1213, 1297, 1303, 1327, 1423, 1471, 1549, 1567, 1693, 1699, 1723, 1747, 1777, 1783, 1801, 1987, 2011, 2053, 2083, 2143, 2161

Offset

1,1

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Example

26 + 6 = 32 == 1 mod 31.
26 * 6 = 156 == 1 mod 31.
26 ^ 6 = 308915776 == 1 mod 31.
so 31 is in the sequence.

Mathematica

okQ[n_] := Block[{x, y, r}, r = Reduce[x+y == 1 && x*y == 1 , {x, y}, Modulus -> n]; r =!= False && Or @@ ((1 == PowerMod[#[[1]], #[[2]], n]) & /@ ({x, y} /. List@ ToRules@ r))]; Select[Prime@Range@300, okQ] (* Giovanni Resta, Feb 03 2014 *)

Prog

(PARI) okp(p) = {for (x=0, p, for (y=0, p, if ((((x+y) % p)==1) && ((x*y) % p == 1) && (((x^y) % p) == 1), print1("x=", x, " y=", y); return (1)); ); ); return (0);
listp(nn) = {forprime (p=2, nn, if (okp(p), print(" p=", p)); ); } \\ Michel Marcus, Feb 02 2014

Crossrefs

Cf. A007645.

Sequence in context: A339545 A102978 A218537 * A222590 A107165 A066811

Adjacent sequences: A236966 A236967 A236968 * A236970 A236971 A236972

Keyword

nonn

Author

Jon Perry, Feb 02 2014

Status

approved