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

0, 2, 3, 7, 8, 16, 26, 5, 21, 18, 38, 49, 50, 16, 26, 6, 81, 54, 98, 70, 157, 161, 58, 147, 21, 86, 92, 197, 50, 249, 137, 184, 119, 45, 45, 261, 198, 61, 176, 143, 51, 103, 221, 72, 11, 219, 35, 86, 385, 384, 141, 143, 225, 92, 245, 533, 557, 473, 170, 375, 516

Offset

1,2

Comments

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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.

Example

a(9) = 21, since 47 is the ninth term of A045309 and 21 is the only solution mod 47 of x^3 = 2.

Maple

Res:=0, 2: count:= 2: p:= 3:
while count < 100 do
p:= nextprime(p);
   if p mod 3 = 2 then
    count:= count+1;
    Res:= Res, numtheory:-mroot(2, 3, p);
fi
od:
Res; # Robert Israel, Sep 12 2018

Mathematica

terms = 100;
A045309 = Select[Prime[Range[2 terms]], Mod[#, 3] != 1&];
a[n_] := PowerMod[2, 1/3, A045309[[n]]];
Array[a, terms] (* Jean-François Alcover, Feb 27 2019 *)

Crossrefs

Cf. A040028, A045309, A059940, A060914, A060122, A060123, A060124.

Sequence in context: A076550 A062269 A244508 * A002964 A166966 A247843

Adjacent sequences: A060118 A060119 A060120 * A060122 A060123 A060124

Keyword

nonn

Author

Klaus Brockhaus, Mar 02 2001

Status

approved