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1:49 PM
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Q: Possible differences of integer sequences and their permutations

Smiley1000Given a sequence $(a_n)_{n \in \mathbb{N}}$, what are necessary and sufficient conditions on $(a_n)_{n \in \mathbb{N}}$ so that there exists a sequence $(b_n)_{n \in \mathbb{N}}$ and a bijection $\sigma: \mathbb{N} \to \mathbb{N}$ with $\forall n \in \mathbb{N}: a_n = b_n - b_{\sigma(n)}$? Additi...

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2 hours later…
3:25 PM
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Q: A pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ such that $f$ and $g$ are idempotent, commute with each other and $f \times g$ is bijective

Smiley1000The question The question is: Does there exist a pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ satisfying the following properties? $f$ and $g$ are idempotent, meaning that $\forall n \in \mathbb{N}$: $f(f(n)) = f(n)$ and $\forall n \in \mathbb{N}$: $g(g(n)) = g(n)$ $f$ and $g$ commute, me...

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