{"paper":{"title":"The generalized Pillai equation $\\pm r a^x \\pm s b^y = c$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Reese Scott, Robert Styer","submitted_at":"2011-02-22T03:45:07Z","abstract_excerpt":"In this paper we consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c$ in nonnegative integers $x, y$ and integers $u, v \\in \\{0,1\\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. We show that $N \\le 2$ when $\\gcd(ra, sb) =1$ and $\\min(x,y)>0$, except for a finite number of cases that can be found in a finite number of steps. For arbitrary $\\gcd(ra, sb)$ and $\\min(x,y) \\ge 0$, we show that when $(u,v) = (0,1)$ we have $N \\le 3$, with an infinite number of cases for which N=3."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4834","kind":"arxiv","version":1},"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"}