How to prove or disprove that there exists a set of natural numbers $a_{1}
An increasing sequence $a_i \in \mathbb{N}$ such that $a_{j}-a_{i}\mid a_{j}-1$ for all i
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number-theory
1 Answers
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There is no such sequence.
Suppose by contradiction that we have such a sequence, and for any $j>i$ we have $a_j-a_i\vert a_j-1$. Then either $a_j=1$ or $a_i=1$ or $a_j>2(a_j-a_i)$ (because $a_j=1+k(a_j-a_i)$ for some $k\geq 0$, the three cases corresponding to $k=0,1$ and $k\geq 2$).
The first two cases are only possible for small $i$, so we can discard them (we can assume that the sequence starts with arbitrarily large numbers by cutting off the beginning).
The last one gives us $2a_i>a_j$ with any sufficiently large fixed $i$ and all $j>i$. But of course $a_{i+a_i}\geq 2a_i$, so we have a contradiction.