R. Ernvall & T. Metsänkylä: Tables of vanishing Fermat quotients

These are tables of the pairs (p,k) such that the Fermat quotient q(k) = (k^{p-1}-1)/p vanishes mod p. Here p is an odd prime and k is an integer prime to p. The tables cover the primes p up to one million and, for each prime, the range 1 < k < p.

The tables were computed in 1995 and their contents were analyzed and summarized in the authors' article 'On the p-divisibility of Fermat quotients', Math. Comp. 66 (1997), 1353-1365.

The tables are numbered from 1 to 10. Table Nr. n covers the range (n-1) x 10^5 < p < n x 10^5.

A typical line in the tables is of the form

with each [] being a string of the form k = p_1.p_2. ... p_s. Here k denotes a number in the range 1 < k < p such that q(k) = 0 mod p, and p_1.p_2. ... p_s is the prime factorization of k. The last entry #r gives the total number of such k, including the trivial value k = 1 (thus, the number of []'s is r-1).
See Table1 as an example.

Only those primes p appear in the tables for which r > 1, i.e., the list of []'s is not empty.

At the end of each table there is a summary giving the distribution of primes p according to the values of r, i.e., the total number N_r of p belonging to a fixed r (including r = 1). The bottom line of the summary shows the sum of N_r (equal to the total number of odd primes contained in the interval in question).

Retrieve individual tables by clicking corresponding items in the following table.