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%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