The function ${\sum }_{p\le x}1/p-loglog\left(x\right)-M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

We completely solve the Diophantine equations $x²+2^a{q}^b=yⁿ$ (for q = 17, 29, 41). We also determine all $C=p{₁}^{a₁}\cdots p_k^{a_k}$ and $C=2^{a₀}p{₁}^{a₁}\cdots p_k^{a_k}$, where $p₁,...,p_k$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

We prove that the sums $S_k$ of independent random vectors satisfy $P(ma{x}_{1\le k\le n}\parallel S_k\parallel >3t)\le 2ma{x}_{1\le k\le n}P(\parallel S_k\parallel >t)$, t ≥ 0.

We first investigate factorizations of elements of the semigroup $S$ of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for $\rho \left(S\right)$, and, given $A\in S$, also provide formulas for $l\left(A\right)$, $L\left(A\right)$ and $\rho \left(A\right)$. As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem...