Karl Dilcher

A search for Wieferich and Wilson primes

An odd prime p is called a Wieferich prime if 2 P-1 = 1 (mod p 2 ) alternatively, a Wilson prime if (p - 1)|= -1 (mod p 2 ). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5,13, and 563. We report that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson primes p < 5x 10 8 . It is elementary that both defining congruences above hold merely (mod p), and it is sometimes estimated on heuristic grounds that the probability that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p).

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ ikxk. In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤ β (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each β) for which the independence polynomials have a root arbitrarily close to −β.

Some q -series identities related to divisor functions

Abstract The generating functions of the divisor functions σk(n) = Σd|ndk are expressed as sums of products of the series U m (q):= ∑ n=1 ∞ n m q n ∏ j=n+1 ∞ (1−q j ), m=1,…,k+1 , and vice versa. Other related q-series identities are derived, including ∑ n=k ∞ n k (1−q j = ∑ j 1 =1 ∞ q j 1 1−q j 1 ∑ j 2 =1 j 1 q j 2 1−q j 2 … ∑ j k =1 j k−1 q j k 1−q j k .

Pi, Euler Numbers, and Asymptotic Expansions

Gregory’s series for π, truncated at 500,000 terms, gives to forty places

Shortened recurrence relations for Bernoulli numbers

4\sum\limits_{k = 1}^{500.000} {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{2k - 1}}} = 3.141590653589793240462643383269502884197

Resultants and discriminants of Chebyshev and related polynomials

The work of Lehmer and Briggs on Euler constants in arithmetical progressions is extended to the generalized Euler constants that arise in the Laurent expansion of g(s) about s = 1 . The results are applied to the summation of several classes of slowly converging series. A table of the constants is provided.

Wilson quotients for composite moduli

Starting with two little-known results of Saalschutz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.