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.

First, some classic properties of a weighted Frobenius-Perron operator ${𝒫}_{\varphi }^u$ on $L^1\left(\Sigma \right)$ as a predual of weighted Koopman operator $W=u{U}_{\varphi }$ on $L^{\infty }\left(\Sigma \right)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of ${𝒫}_{\varphi }^u$ under certain conditions.

It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm and what is its time compexity. The technical hitch is in fixing the right sign of the square root and this is the heart of the discrete logarithm problem over finite fields of characteristic not equal to 2. In this paper a couple of probabilistic algorithms to compute the discrete...