2,1

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^3 = 2 iff n^3-2 has a prime factor > n; n is a solution mod p of x^3 = 2 iff p is a prime factor of n^3-2 and p > n.

n^3-2 has at most two prime factors > n, consequently these factors are the only primes p such that n is a solution mod p of x^3 = 2. For n such that n^3-2 has no prime factor > n (the zeros in the sequence; they occur beyond the last entry shown in the database) see A060591. For n such that n^3-2 has two prime factors > n, cf. A060914.

If n^3-2 has prime factors > n, then a(n) = least of these prime factors, else a(n) = 0.

a(2) = 3, since 2 is a solution mod 3 of x^3 = 2, and 2 is not a solution mod p of x^3 = 2 for prime p = 2. Although 2^3 = 2 mod 2, prime 2 is excluded because 0 < 2 and 2 = 0 mod 2. a(5) = 41, since 5 is a solution mod 41 of x^3 = 2, and 5 is not a solution mod p of x^3 = 2 for primes p < 41. Although 5^3 = 2 mod 3, prime 3 is excluded because 3 < 5 and 5 = 2 mod 3.

Cf. A040028, A060121, A060122, A060123, A060124, A060591, A060914.

Adjacent sequences: A059937 A059938 A059939 this_sequence A059941 A059942 A059943

Sequence in context: A100856 A100858 A035410 this_sequence A106893 A047105 A136361

nonn

Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 02 2001