We prove two conjectural congruences on the $(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.

Let $q\geq 1$ be any integer and let $\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes $p$ satisfying

$$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$

$g$

$p$

$\text{gcd}(g,(p-1)/q)=1$

We establish some supercongruences for the truncated $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series involving the $p$-adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated $_{3}F_{2}$ hypergeometric series. Related supercongruences modulo $p^{3}$ are proposed as conjectures.

We provide two new bounds on the number of visible points on exponential curves modulo a prime for all choices of primes. We also provide one new bound on the number of visible points on exponential curves modulo a prime for almost all primes.

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross–Koblitz formula. In this paper we prove four supercongruences between two truncated hypergeometric series $_{2}F_{1}$. The results generalise the four Rodriguez-Villegas supercongruences.

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

A terrace for m is an arrangement (a1, a2, . . . , am) of the m elements of m such that the sets of differences ai+1 − ai and ai − ai+1 (i = 1, 2, . . . , m − 1) between them contain each element of m \ {0} exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for m; each such terrace may be partitioned into segments, one of which contains merely the zero element of m, whereas each other segment is either (a) a sequence of successive powers of an element of m or (b) such a sequence multiplied throughout by a constant. We now adapt this idea by using power-sequences in n, where n is an odd prime power, to obtain terraces for m, where m = n − 2. We write each element from n so that they lie in the interval [0, n − 1] and then delete 0 and n − 1 so that they leave n − 2 elements that may be interpreted as the elements of n−2. A segment of one of the new terraces may be of type (a) or (b), incorporating successive powers of 2, with each entry evaluated modulo n. Our constructions provide n−2 terraces for all odd primes n satisfying 0 < n < 1,000 except for n = 127, 241, 257, 337, 431, 601, 631, 673, 683, 911, 937 and 953. We also provide n−2 terraces for n = 3r (r > 1) and for some values n = p2, where p is prime.

For a large integer $m$, we obtain asymptotic formulae for the number of solutions of certain congruences modulo $m$ with several variables, where the variables belong to special sets of residue classes modulo $m$. In particular, we obtain new information on the exceptional set of the multiplication table problem in the residue ring modulo $m$.