Some Problems In Number Theory

In this paper, we consider seven problems in number theory:the decomposition of triangular numbers, perfect numbers and Fibonacci primes, a new variant of the Hilbert-Waring problem, Mordell curve and arithmetic sequence of order two, a generalization of Wolstenholme’s theorem, Erdos-Straus conjecture and two special cases, a new generalization of Fermat’s Last Theorem. We obtain several results as follow:1. There are infinitely many triangular numbers which have two different ways to be decomposed as the product of two triangular numbers, each greater than1.2. We introduce the concept of F-perfect number, which is a positive integer n such that∑d|n,d<n d2=3n.And prove that all the F-perfect numbers are of the form n=F2k-1F2k+1, where both F2k-1and F2k+1are Fibonacci primes.3. We propose a new variant of Waring’s problem:to express a positive integer n as a sum of s positive integers whose product is a k-th power. We define, in a similar way to that done for g(k) and G(k) in Waring’s problem, numbers g’(k) and G’(k). We obtain g’(k)=2k-1, G’(p)≤p+1for primes p, G’(m)≤m+2for composite numbers m. Moreover, we make two conjectures about G’(3) and G’(4).4. We consider an arithmetic property about the integral points of Mordell curve:the most number of consecutive squares or triangular numbers included in y-coordinate of integral points. We prove that there are at most three consecutive triangular numbers; and conjecture that there are at most two consecutive squares.5. We obtain a generalization of Wolstenholme’s theorem.6. We give a new necessary condition for Erdos-Straus conjecture and discuss two special cases of Erdos-Straus conjecture.7. We consider a generalization of Fermat’s Last Theorem:A+B=C such that ABC=Dn. We discuss several cases for gcd(A, B,C)=pk where p is an odd prime. In particular. for k=1we prove that it has no nonzero integer solutions when n=3and we conjecture that it is also true for any prime n>3.