9 research outputs found
Zero-sum problems for abelian p-groups and covers of the integers by residue classes
Zero-sum problems for abelian groups and covers of the integers by residue
classes, are two different active topics initiated by P. Erdos more than 40
years ago and investigated by many researchers separately since then. In an
earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60],
the author claimed some surprising connections among these seemingly unrelated
fascinating areas. In this paper we establish further connections between
zero-sum problems for abelian p-groups and covers of the integers. For example,
we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If
{a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly
2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there
exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in
I}c_s=0. Our main theorem in this paper unifies many results in the two realms
and also implies an extension of the Alon-Friedland-Kalai result on regular
subgraphs
Mixed sums of primes and other terms
In this paper we study mixed sums of primes and linear recurrences. We show
that if m=2(mod 4) and m+1 is a prime then
for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer,
u_0=0, u_1=1 and u_{i+1}=au_i+u_{i-1} for i=1,2,3,..., then all the sums
u_m+au_n (m,n=1,2,3,...) are distinct. One of our conjectures states that any
integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number
and a positive Fibonacci number.Comment: 11 page