Abstract
The goal of the paper is to consider a special class of combinatorial problems, the solution of which is realized by constructing finite sequences of \(\pm 1\). For example, for fixed \(p\in {\mathbb N}\), is well known the existence of \(n_p\in {\mathbb N}\) with the property: any set of \(n_p\) consecutive natural numbers can be divided into 2 sets, with equal sums of its pth-powers. The considered property remains valid also for sets of finite arithmetic progressions of integers, real or complex numbers. The main observation here is the generalization of the results for arithmetic progressions with elements of complex field \({\mathbb C}\) to elements of arbitrary associative, commutative algebra.
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Notes
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This paper is continuation of Stoenchev and Todorov (2022), and contains complete proofs of the formulated propositions.
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Acknowledgements
Miroslav Stoenchev is supported by Bulgarian National Science Fund under Project KP-06-M32/2—17.12.2019 “Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics”. Venelin Todorov is supported by the Bulgarian National Science Fund under Projects KP-06-N52/5 “Efficient methods for modeling, optimization and decision making” and by the Bilateral Project KP-06-Russia/17 “New Highly Efficient Stochastic Simulation Methods and Applications”.
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Stoenchev, M., Todorov, V. (2022). Combinatorial Etudes and Number Theory. In: Fidanova, S. (eds) Recent Advances in Computational Optimization. WCO 2021. Studies in Computational Intelligence, vol 1044. Springer, Cham. https://doi.org/10.1007/978-3-031-06839-3_12
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