(To get these irregular pairs, we only need to factorize a n. The following table shows all irregular pairs with n ≤ 63. ( Weak irregular index is defined as "number of integers 0 ≤ n ≤ p − 2 such that p divides a n.) Besides, 491 is the only prime below 1000 with weak irregular index 4, and all other odd primes below 1000 with weak irregular index 0, 1, or 2. The following table shows all irregular pairs with odd prime p ≤ 661:
Thus, the definition of irregular pair ( p, n), n should be at most p − 2. For example, since 19 divides a 11 and 2 × 19 = 38 does not divide 11, so 19 divides a 18 k+11 for all k. If p is an irregular prime and p divides the numerator of the Bernoulli number B 2 k for 0 1). Although the two latter tables did not appear in print, Johnson found that ( p, p − 3) is in fact an irregular pair for p = 16843 and that this is the first and only time this occurs for p 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1, and later generalized it. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Kummer found the irregular primes less than 165. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either ( p, p − 3) or ( p, p − 5) fails to be an irregular pair. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if ( p, p − 3) is not an irregular pair. This focused attention on the irregular primes. In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p is regular.
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