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I wish I had a real answer for you! You are essentially interested in a tough conjecture of Hartmann, known as the "halving conjecture", which is promoted heavily by Reza Khosrovshahi. Actually, the …

This is barely of research interest, but I'd classify it as a curiosity with connections to combinatorics. The problem is to place integers in an $n \times n \times n$ array so that all $3n^2$ line s …

Nice question. This is not (any longer) an answer, but a strategy. First, try to construct 25 mutually disjoint squared squares of the same order. Then arrange them according to a 3,4,5 template. …

I guess the answer should be that $(a,b,c)$ fills the line if and only if there is an integral combination of its permutations equaling $(1,1,1)$. We want the elementary divisors of a certain $3 \tim …

Let $a,b,c$ be positive integers with gcd$(a,b,c)=1$, and let $\mathbb{N}$ denote the set of nonnegative integers. It is well known that $\mathbb{N} \setminus (a \mathbb{N}+b \mathbb{N} + c \mathbb{N …

This is probably quite naïve, maybe even stackexchange-worthy. Consider a quadratic form such as $Q(x,y) = 3x^2+y^2$. We know that, for primes $p \equiv 1 \pmod{3}$, there exist integer solutions to …