Suppose I have a sequence $a_n$, whose entries are the ordered elements of $S_{x,y}$: $S_{x, y}= \{ z \mid \left( z=n_1x+n_2y \right) \wedge \left( n_1, n_2 \in \mathbb{N}_1 \right) \wedge \left( \gcd \left( n_1, n_2 \right) = 1 \right)\}$ $x, y \in \mathbb{N}_1$ The Question: Given that $\gcd \left( x, y \right) = 1$, is […]

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. Batominovski showed that this conjecture is indeed true and the maximum unrepresentable natural number is $O((xy)^{1+ε})$ for any $ε > 0$. But an obvious […]

Let $p$ be a positive integer. For each nonnegative integer $k$, write $[k]$ for the set $\{0,1,2,\ldots,k\}$. Also, we define $[-1]:=\emptyset$. We say that an integer $k\geq -1$ is $p$-splittable if there is a partition of $[k]$ into $p$ subsets $A_1$, $A_2$, $\ldots$, $A_p$ such that $\sum_{x\in A_1}\,x=\sum_{x\in A_2}\,x=\ldots=\sum_{x\in A_p}\,x$ (i.e., these sets have the […]

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