An odd prime p is called a Wieferich prime if 2p−1 ≡ 1 (mod p); alternatively, a Wilson prime if (p− 1)! ≡ −1 (mod p). 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 < 4×1012 , and no new Wilson primes p < 5×108. It is elementary that both defining congruences above h...

Let ψm be the smallest strong pseudoprime to the first m prime bases. This value is known for 1 ≤ m ≤ 11. We extend this by finding ψ12 and ψ13. We also present an algorithm to find all integers n ≤ B that are strong pseudoprimes to the first m prime bases; with a reasonable heuristic assumption we can show that it takes at most B2/3+o(1) time.

We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estim...

The two tissue-specific mRNAs encoding the isozymes of rabbit angiotensin-converting enzyme (ACE) are generated from the same gene by alternative choice of two transcription initiation sites 5.7 kb apart. In the current study, we have characterized the regulatory sites controlling the transcription of the larger pulmonary isozyme mRNA. For this purpose, reporter genes driven by varying lengths ...

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