The result of the distributed computing projectWieferich@Home is presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 obtained by replication of a bit string of bit pseudo-length k ≤ 24 and increased by one are Wieferich primes only for the cases of 1092 or 3510.

This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime $p$ is Fibonacci-Wieferich is equal to $1/p$. According to our computational results and some theoretical consideratons, another form of probability can...

We explain the algorithms that we have implemented to show that all integers congruent to $4$ modulo $80$ in the interval $[6\times {10}^{12};2.17\times {10}^{14}]$ are sums of five fourth powers, and that all integers congruent to $6,21$ or $36$ modulo $80$ in the interval $[6\times {10}^{12};1.36\times {10}^{23}]$ are sums of seven fourth powers. We also give some results related to small sums of biquadrates. Combining with the Dickson ascent method, we deduce that all integers in the interval $[13793;{10}^{245}]$ are sums of $16$ biquadrates.