On some Diophantine equations. II.
On some Diophantine equations. III.
On Some Elliptic Curves Defined Over ... of Free Rank ... 9.
On some elliptic curves with large sha.
On some equations over finite fields
In this paper, following L. Carlitz we consider some special equations of variables over the finite field of elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on and .
On some expressions associated with Fermat's last theorem.
On some generalizations of the diophantine equation
On some generalizations of the diophantine equation
On some inequalities connected with Fermat's equation.
On some properties of two simultaneous polygonal sequences.
On some special forms of simultaneous Pell equations
On Some Ternary Quartic Diophantine Equations.
On some universal sums of generalized polygonal numbers
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in...
On Sophie Germain type criteria for Fermat's Last Theorem
On square values of quadratics
On squares of squares
On squares of squares II
On sums and products in a field
We study sums and products in a field. Let be a field with , where is the characteristic of . For any integer , we show that any can be written as with and , and that for any we can write every as with and . We also prove that for any and there are such that .
On sums of powers and a related problem
On sums of -units and linear recurrences