A236969 - OEIS

A236969

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

%I #25 Nov 06 2024 04:11:12

%S 3,19,31,61,67,79,97,127,307,331,367,409,457,499,571,613,631,691,709,

%T 727,733,787,829,883,991,1063,1087,1093,1213,1297,1303,1327,1423,1471,

%U 1549,1567,1693,1699,1723,1747,1777,1783,1801,1987,2011,2053,2083,2143,2161

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

%H Giovanni Resta, <a href="/A236969/b236969.txt">Table of n, a(n) for n = 1..1000</a>

%e 26 + 6 = 32 == 1 mod 31.

%e 26 * 6 = 156 == 1 mod 31.

%e 26 ^ 6 = 308915776 == 1 mod 31.

%e so 31 is in the sequence.

%t 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 *)

%o (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);}

%o listp(nn) = {forprime (p=2, nn, if (okp(p), print(" p=", p)););} \\ _Michel Marcus_, Feb 02 2014

%Y Cf. A007645.

%K nonn

%O 1,1

%A _Jon Perry_, Feb 02 2014