Let p > 2 be a prime number. For each integer 0 < n < p, define n by the congruence nn ≡ 1 (mod p) with 0 < n < p. We are led to study the distribution behavior of n− n+ 1 in order to prove the asymptotic formula p−2 ∑ n=1 |n− n+ 1| = 1 3 p +O(p log p).

Suppose a 1 < a 2 < : : : < a Z are distinct integers in f1; : : :; N g. If a 0 +" 1 a 1 +" 2 a 2 + + " Z a Z is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N. The argument uses a modiied form of Gallagher's Larger Sieve, and shows that if Z > (9=2 + o(1)) log N= log log N , then for some prime p < Z 2 , the subset sums " 1 a 1 + " 2 a 2 + + " Z...

This work studies the existence of positive prime periodic solutions of higher order for rational recursive equations of the form yn = A + yn−1 yn−m , n = 0, 1, 2, . . . , with y−m , y−m+1, . . . , y−1 ∈ (0,∞) and m ∈ {2, 3, 4, . . .}. In particular, we show that for sufficiently small A > 0, there exist periodic solutions with prime period 2m +Um + 1, for almost all m, where Um = max{i ∈ N : i...

The main combinatorial result in this article is a classification of bar partitions of n which are of maximal p-bar weight for all odd primes p ≤ n. As a consequence, we show that apart from very few exceptions any irreducible spin character of the double covers of the symmetric and alternating groups vanishes on some element of odd prime order.

The sequence of (terminating balanced) hypergeometric sums an = n ∑ k=0 ( n k )2( n+ k k )2 , n = 0, 1, . . . , appears in Apéry’s proof of the irrationality of ζ(3). Another example of hypergeometric use in irrationality problems is Ramanujan-type identities for 1/π, like ∞ ∑ k=0 ( 2k k )3 (4k + 1) (−1) 26k = 2 π . These two, seemingly unrelated but both beautiful enough, hypergeometric series...

In this paper, a constructive general matrix factorization scheme is developed for extracting a nontrivial factor from a given nD (n > 2) polynomial matrix whose maximal order minors satisfy certain conditions. It is shown that three classes of nD polynomial matrices admit this kind of general matrix factorization. It turns out that minor prime factorization and determinantal factorization are ...

The Invariant Maximal Clique Theorem asserts that every graph on Q[0,n] with a specific invariance condition has a maximal clique with another specific invariance condition. Here Q[0,n] consists of the rationals in the interval [0,n]. The invariance conditions are all given by equivalence relations involving only < on [0,n] and the distinguished elements 1,...,n. We prove the Invariant Maximal ...

The Invariant Maximal Clique Theorem asserts that every graph on Q[0,n] with a specific invariance condition has a maximal clique with another specific invariance condition. Here Q[0,n] consists of the rationals in the interval [0,n]. The invariance conditions are all given by equivalence relations involving only < on [0,n] and the distinguished elements 1,...,n. We prove the Invariant Maximal ...

Let G = GLn(q) be the general linear group of degree n ≥ 2 defined over a finite field Fq of characteristic p. We fix a prime l 6= p and let stand R for a local principal ideal domain having characteristic 0, maximal ideal lR, and containing a primitive p-th root of unity. Then the residue field K = R/lR has characteristic l and a primitive p-th root of unity. By a Steinberg lattice of G over R...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...