Congruences for m-ary partitions.
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 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.
We employ recent results on Jacobi forms to investigate congruences and filtrations of Siegel modular forms of degree . In particular, we determine when an analog of Atkin’s -operator applied to a Siegel modular form of degree is nonzero modulo a prime . Furthermore, we discuss explicit examples to illustrate our results.
A prime is said to be a Wolstenholme prime if it satisfies the congruence . For such a prime , we establish an expression for given in terms of the sums (. Further, the expression in this congruence is reduced in terms of the sums (). Using this congruence, we prove that for any Wolstenholme prime we have Moreover, using a recent result of the author, we prove that a prime satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique...
Let be a prime, and let be the Fermat quotient of to base . In this note we prove that 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 which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum modulo that also generalizes a...