Abstract
In this paper, we provide some results concerning the structure of a multiset A with elements from \(\mathbb {Z}_n^{\times }\), which has non-empty subset sums equally distributed modulo n. Here, \(\mathbb {Z}_n^{\times }\) denotes the set which contains all the invertible elements of the ring \(\mathbb {Z}_n\). In particular, we prove that if n belongs to a certain subset of the natural numbers, then A is a union of sets of the form \(\{ a\cdot (\pm 2^i)\}\).
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21 June 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00013-023-01864-z
Notes
This method was used also by Euler himself to prove that \(a^{\varphi (n)}\equiv 1\pmod {n}\).
Here we adopt the convention that the empty sum is \(0\pmod {n}\)
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Konstantinos, G. On subset sums of \(\mathbb {Z}_n^{\times }\) which are equally distributed modulo n. Arch. Math. 121, 47–54 (2023). https://doi.org/10.1007/s00013-023-01853-2
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DOI: https://doi.org/10.1007/s00013-023-01853-2