Let p(n) denote the number of partitions of n. In this paper we prove that if {An + B} is an arithmetic progression and l ≥ 5 a prime, such that p(An+B) ≡ 0 (mod l), n ∈ N. Then l|A and (

In this paper we find solutions of congruences of the form ax ≡ b(mod p), k≥1 where a, b and n>0 are integers which are not divisible by a prime p using Halley's iterative algorithm. An algorithm is also proposed to reduce the degree of a nonlinear congruence modulo p. AMS Mathematics Subject Classification (2000): 05C25 • 11E04 • 20G15

Let $q$ be an odd prime and $B = \{b_{j}\}_{j=1}^{l}$ a finite set of nonzero integers that does not contain perfect $q^{th}$ power. We show $B$ has power modulo every $p \neq q$ dividing $\prod_{b\in B} b$ if only corrresponds to linear hyperplane covering $\mathbb{F}_{q}^{k}$. Here, $k$ is the number distinct factors $q$-free part elements $B$. Consequently: $(i)$ \subset\mathbb{Z}\setminus\{...

Let us assume we are given a symmetric prime b. It is well known that V ∼ 0. We show that Σ ∈ κ. In [16], it is shown that B > π. Recent developments in discrete graph theory [15, 3] have raised the question of whether

Let A: be an integer > 2 andp a prime such that vk(p) = (k,p — 1) > 1. Let bn + c(n = 0,1,. ..;b > 2,1 < c < b, (b,p) — (c,p) = 1) be an arithmetic progression. We denote the smallest kth power nonresidue in the progression bn + c by g(p,k,b,c), the smallest quadratic residue in the progression bn + c by r2(p,b,c), and the nth smallest prime kth power nonresidue by g„(p,k), n = 0, 1, 2,_ If C(p...

Let A and B be unital rings. An additive map T:A→B is called a weighted Jordan homomorphism if c=T(1) an invertible central element cT(x2)=T(x)2 for all x∈A. We provide assumptions, which are in particular fulfilled when A=B=Mn(R) with n≥2 R any ring 12, under every surjective the property that T(x)T(y)+T(y)T(x)=0 whenever xy = yx 0 homomorphism. Further, we show prime char(A)≠2,3,5, then bijec...