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

p : []. []. ... []. #r

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.

Table1 Table2 Table3 Table4 Table5
Table6 Table7 Table8 Table9 Table10