A059940 Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.

3, 5, 31, 41, 107, 11, 17, 727, 499, 443, 863, 439, 457, 3373, 23, 1637, 53, 6857, 31, 47, 5323, 811, 6911, 919, 29, 19681, 439, 739, 13499, 29789, 43, 7187, 43, 461, 23327, 50651, 59, 2579, 2909, 22973, 2179, 15901, 14197, 293, 1187, 34607, 11059

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..48.

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A189739 A035410 A290487 * A106893 A175097 A217320

Adjacent sequences: A059937 A059938 A059939 * A059941 A059942 A059943

Keyword

nonn

Author

Klaus Brockhaus, Mar 02 2001

Status

approved