{"paper":{"title":"Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A. Oller-Marcen, Catalina Calderon, Jose Maria Grau, L\\'aszl\\'o T\\'oth","submitted_at":"2014-03-31T06:02:05Z","abstract_excerpt":"In this paper we introduce and study a family $\\Phi_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\\gcd(x_1^2+\\ldots +x_k^2, n)=1$ with $x_1,\\ldots,x_k \\in {\\mathbb{Z}}/n{\\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\\mathbb{Z}}/n{\\mathbb{Z}}$. We prove that $\\Phi_k$ is multiplicative for every $k$, we obtain an explicit formula for $\\Phi_k(n)$ in terms of the prime-power decomposition of $n$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7878","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}