Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]
In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]
Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]