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### A note on Sierpiński's problem related to triangular numbers

Colloquium Mathematicae

We show that the system of equations ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r}$, where ${t}_{x}=x\left(x+1\right)/2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r},{t}_{x}+{t}_{y}+{t}_{z}={t}_{s}$ has infinitely many rational two-parameter solutions.

### A note on the equation ${\left(x+y+z\right)}^{2}=xyz$.

General Mathematics

### A propos du problème de Waring

Seminaire de Théorie des Nombres de Bordeaux

Acta Arithmetica

### A variety of Euler's sum of powers conjecture

Czechoslovak Mathematical Journal

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $\left\{\begin{array}{c}n={a}_{1}+{a}_{2}+\cdots +{a}_{s-1},\hfill \\ {a}_{1}{a}_{2}\cdots {a}_{s-1}\left({a}_{1}+{a}_{2}+\cdots +{a}_{s-1}\right)={b}^{s}\hfill \end{array}\right\$ has positive integer or rational solutions $n$, $b$, ${a}_{i}$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...

Acta Arithmetica

### An identity Ramanujan probably missed

Colloquium Mathematicae

"Ramanujan's 6-10-8 identity" inspired Hirschhorn to formulate his "3-7-5 identity". Now, we give a new "6-14-10 identity" which we suppose Ramanujan would have discovered but missed to mention in his notebooks.

Acta Arithmetica

### Bihomogeneous forms in many variables

Journal de Théorie des Nombres de Bordeaux

We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Bounds of prime solutions of some diagonal equations.

Journal für die reine und angewandte Mathematik

### Cubic moments of Fourier coefficients and pairs of diagonal quartic forms

Journal of the European Mathematical Society

We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.

Acta Arithmetica

Acta Arithmetica

### Diophantine equation $\frac{{q}^{n}-1}{q-1}=y$ for four prime divisors of $y-1$

Commentationes Mathematicae Universitatis Carolinae

In this paper the special diophantine equation $\frac{{q}^{n}-1}{q-1}=y$ with integer coefficients is discussed and integer solutions are sought. This equation is solved completely just for four prime divisors of $y-1$.

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