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A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

%I A060121 #10 Feb 27 2019 16:16:24
%S A060121 0,2,3,7,8,16,26,5,21,18,38,49,50,16,26,6,81,54,98,70,157,161,58,147,
%T A060121 21,86,92,197,50,249,137,184,119,45,45,261,198,61,176,143,51,103,221,
%U A060121 72,11,219,35,86,385,384,141,143,225,92,245,533,557,473,170,375,516
%N A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.
%C A060121 Solutions mod p are represented by integers from 0 to p-1. For i > 1, i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3-2 and p > i (cf. comment to A059940). i^3-2 has at most two prime factors > i. Hence i is a solution mod p of x^3 = 2 for at most two different p and therefore no integer occurs more than twice in this sequence. There are integers which do occur twice, e.g. 16, 21, 26 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.
%H A060121 Robert Israel, <a href="/A060121/b060121.txt">Table of n, a(n) for n = 1..10000</a>
%F A060121 a(n) = first (only) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has only one solution mod p, i.e. p is the n-th term of A045309.
%e A060121 a(9) = 21, since 47 is the ninth term of A045309 and 21 is the only solution mod 47 of x^3 = 2.
%p A060121 Res:=0,2: count:= 2: p:= 3:
%p A060121 while count < 100 do
%p A060121 p:= nextprime(p);
%p A060121    if p mod 3 = 2 then
%p A060121     count:= count+1;
%p A060121     Res:= Res, numtheory:-mroot(2,3,p);
%p A060121 fi
%p A060121 od:
%p A060121 Res; # _Robert Israel_, Sep 12 2018
%t A060121 terms = 100;
%t A060121 A045309 = Select[Prime[Range[2 terms]], Mod[#, 3] != 1&];
%t A060121 a[n_] := PowerMod[2, 1/3, A045309[[n]]];
%t A060121 Array[a, terms] (* _Jean-François Alcover_, Feb 27 2019 *)
%Y A060121 Cf. A040028, A045309, A059940, A060914, A060122, A060123, A060124.
%K A060121 nonn
%O A060121 1,2
%A A060121 _Klaus Brockhaus_, Mar 02 2001