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A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists. 6
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A076550 A062269 A244508 * A002964 A166966 A247843
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Mar 02 2001
STATUS
approved

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Last modified May 23 18:34 EDT 2024. Contains 372765 sequences. (Running on oeis4.)