Representations of Numbers as Sums and Differences of Unlike Powers

Jabara, Enrico

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 1, pp. 169-177

Résumé

In this paper we prove that every $n \in \mathbf{Z}$ can be written as $$n=\epsilon_{1}x^{2}_{1} + \epsilon_{2}x^{3}_{2} + \epsilon_{3}x^{4}_{3} + \epsilon_{4}x^{5}_{4}$$ and as $$n=\epsilon_{1}x^{3}_{1} + \epsilon_{2}x^{4}_{2} + \epsilon_{3}x^{5}_{3} + \epsilon_{4}x^{6}_{4} + \epsilon_{5}x^{7}_{5} + \epsilon_{6}x^{8}_{6} + \epsilon_{7}x^{9}_{7} + \epsilon_{8}x^{10}_{8}$$ with $x_{i} \in \mathbf{Z}$ and $\epsilon_{i} \in \{-1,1\}$. We also prove some other results on numbers expressible as sums or differences of unlike powers.