The numbers 242206083 · 2 38880 ± 1 are twin primes. The number p = 2375063906985 · 2 19380 − 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515 · 2 5056 − 1, the numbers p, p + 2 and 2p + 1 are all primes. In the first days of October, 1995, Harvey Dubner  found the largest known twin primes with 5129 decimal digits.… (More)
In this article, we precise two results of Bagchi, the first one on the universality of the Riemann zeta function the second on its relation with the Riemann hypothesis.
We describe the subgroups of the group Zm × Zn × Zr and derive a simple formula for the total number s(m, n, r) of the subgroups, where m, n, r are arbitrary positive integers. An asymptotic formula for the function n → s(n, n, n) is also deduced.
With the decreasing feature size of today's nanoelectronic circuits , the susceptibility to transient failures increases. New robust and self-adaptive designs are developed, which can handle transient error to some extent, but at the same time make testing for permanent faults more difficult. This paper reviews the " signature rollback " scheme as a… (More)