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 $t_x=x(x+1)/2$ 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.

S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x=a$, $min(x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $gcd(x,y)=1$.

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ 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...

"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.

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.

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.