We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.

This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

Let $F_n$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$:1) ${inf}_{n\in \mathbb{N}}\left|\right|F_n\alpha \left|\right|\le \frac{\varphi -1}{\varphi +2}$,2) ${lim\; inf}_{n\to \infty }\left|\right|F_n\alpha \left|\right|\le \frac{1}{5}$,3) ${lim\; inf}_{n\to \infty }\left|\right|{\varphi }^n\alpha \left|\right|\le \frac{1}{5}$.These results are the best possible.

In this paper the special diophantine equation $\frac{q^n-1}{q-1}=y$ with integer coefficients is discussed and integer solutions are sought. This equation is solved completely just for four prime divisors of $y-1$.

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and ${\mu }_i\in -1,1(i=1,2)$, and let $h(-2^{1-e}D)(e=0or1)$ denote the class number of the imaginary quadratic field $\mathbb{Q}\left(\surd (-2^{1-e}D)\right)$. In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then $h(-2^{1-e}D)\equiv 0\left(modn\right)$, where D, and n satisfy $kⁿ-2^{e+1}=Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

We study the Diophantine equations ${(k!)}^n-k^n={(n!)}^k-n^k$ and ${(k!)}^n+k^n={(n!)}^k+n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.