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In Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML the author has studied the Diophantine equation \begin{equation} x^3+y^...

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...

For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$. Can such a graph be integral, i.e. have only integer eigenvalues? It is easy to see that the ...

Given the roots $x_i$ of the depressed cubic, $$x^3+px+q=0$$ with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that, $$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...