Abstract
Fox and Kleitman proved in 2006 that for any positive integer b, the 2n-variable equation \(x_1+\cdots +x_n - x_{n+1}- \cdots - x_{2n} \ = \ b\) is not 2n-regular. Moreover, they conjectured the existence of an integer \(b_n \ge 1\) such that for \(b=b_n\), this equation is \((2n-1)\)-regular. In this note, we settle the first nontrivial case of the conjecture, namely for \(n=2\), and we propose a slight refinement of it.
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Das Adhikari, S., Eliahou, S. (2017). On a Conjecture of Fox and Kleitman on the Degree of Regularity of a Certain Linear Equation. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_1
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DOI: https://doi.org/10.1007/978-3-319-68032-3_1
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