We systematically investigate the expressions and congruences for both a one-parameter family $\left\{G_n\left(x\right)\right\}$ as well as a two-parameter family $\left\{G_n(r,m)\right\}$ of sequences.

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p/8]}\left(modp\right)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1(mod{p}^4)$. For such a prime $p$, we establish an expression for $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)(mod{p}^8)$ given in terms of the sums $R_i:={\sum }_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1-2p\sum _{k=1}^{p-1}\frac{1}{k}-2p^2\sum _{k=1}^{p-1}\frac{1}{k^2}(mod{p}^7).$$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique...

Let $p>3$ be a prime, and let $q_p\left(2\right)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $$\sum _{k=1}^{p-1}\frac{1}{k·2^k}\equiv q_p\left(2\right)-\frac{p{q}_p{\left(2\right)}^2}{2}+\frac{p^2{q}_p{\left(2\right)}^3}{3}-\frac{7}{48}{p}^2{B}_{p-3}(mod{p}^3),$$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $$q_p{\left(2\right)}^3\equiv -3\sum _{k=1}^{p-1}\frac{2^k}{k^3}+\frac{7}{16}\sum _{k=1}^{(p-1)/2}\frac{1}{k^3}(modp),$$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ${\sum }_{k=1}^{p-1}1/(k^2·2^k)$ modulo $p^2$ that also generalizes a...

In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $\left(x\right)={gx+h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps ${\delta }_1,...,{\delta }_m$ form an increasing (resp....

Nous construisons, dans les corps quadratiques réels, une infinité de fractions continues périodiques uniformément bornées, avec une borne qui semble meilleure que celle connue jusqu’ici. Nous faisons cela en partant de développements en fractions continues de la même forme que ceux des réels $\sqrt{n}+n$. Et ceci nous permet d’obtenir de plus qu’il existe une infinité de corps quadratiques contenant une infinité de développements en fractions continues périodiques formées seulement des entiers $1$ et $2$. Nous...