{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T19:08:05Z","timestamp":1649185685510},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T00:00:00Z","timestamp":1490659200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,3,28]]},"abstract":"Abstract<\/jats:title>\n\t\t\t\tLet Bn<\/jats:sub> = {xi<\/jats:sub> \u00b7 xj<\/jats:sub> = xk<\/jats:sub> : i, j, k \u2208 {1, . . . , n}} \u222a {xi<\/jats:sub> + 1 = xk<\/jats:sub> : i, k \u2208 {1, . . . , n}} denote the system of equations in the variables x1<\/jats:sub>, . . . , xn<\/jats:sub>. For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S \u2286 Bn<\/jats:sub> with a unique solution in positive integers x1<\/jats:sub>, . . . , xn<\/jats:sub>, this solution belongs to [1, b]n. Let g(1) = 1, and let g(n + 1) = 22g(n)<\/jats:sup> for every positive integer n. We conjecture that \u03be (n) 6 g(2n) for every positive integer n. We prove: (1) the function \u03be : N \\ {0} \u2192 N \\ {0} is computable in the limit; (2) if a function f : N \\ {0} \u2192 N \\ {0} has a single-fold Diophantine representation, then there exists a positive integer m such that f (n) < \u03be (n) for every integer n > m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x1<\/jats:sub>, . . . , xp<\/jats:sub>) = 0 and returns a positive integer d with the following property: for every positive integers a1<\/jats:sub>, . . . , ap<\/jats:sub>, if the tuple (a1<\/jats:sub>, . . . , ap<\/jats:sub>) solely solves the equation D(x1<\/jats:sub>, . . . , xp<\/jats:sub>) = 0 in positive integers, then a1<\/jats:sub>, . . . , ap<\/jats:sub> 6 d; (4) the conjecture implies that if a set M \u2286 N has a single-fold Diophantine representation, then M is computable; (5) for every integer n > 9, the inequality \u03be (n) < (22n\u22125<\/jats:sup> \u2212 1)2n\u22125<\/jats:sup> + 1 implies that 22n\u22125<\/jats:sup> + 1 is composite. <\/jats:p>","DOI":"10.1515\/comp-2017-0003","type":"journal-article","created":{"date-parts":[[2017,8,4]],"date-time":"2017-08-04T10:00:50Z","timestamp":1501840850000},"page":"17-23","source":"Crossref","is-referenced-by-count":0,"title":["Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?"],"prefix":"10.1515","volume":"7","author":[{"given":"Apoloniusz","family":"Tyszka","sequence":"first","affiliation":[]}],"member":"374","reference":[{"key":"ref121","first-page":"113","article-title":"Conjecturally computable functions which unconditionally do not have any finite - 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