{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T12:23:19Z","timestamp":1781353399248,"version":"3.54.1"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"325","license":[{"start":{"date-parts":[[2021,4,28]],"date-time":"2021-04-28T00:00:00Z","timestamp":1619568000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","award":["PRIN 2017JTLHJR_006 Geometric"],"award-info":[{"award-number":["PRIN 2017JTLHJR_006 Geometric"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","award":["algebraic and analytic methods in arithmetic"],"award-info":[{"award-number":["algebraic and analytic methods in arithmetic"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","award":["PRA_2018_5 Geometric"],"award-info":[{"award-number":["PRA_2018_5 Geometric"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","award":["algebraic and analytic methods in arithmetic (unipi)"],"award-info":[{"award-number":["algebraic and analytic methods in arithmetic (unipi)"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007514","name":"Universit\u00c3 di Pisa","doi-asserted-by":"publisher","award":["PRIN 2017JTLHJR_006 Geometric"],"award-info":[{"award-number":["PRIN 2017JTLHJR_006 Geometric"]}],"id":[{"id":"10.13039\/501100007514","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007514","name":"Universit\u00c3 di Pisa","doi-asserted-by":"publisher","award":["algebraic and analytic methods in arithmetic"],"award-info":[{"award-number":["algebraic and analytic methods in arithmetic"]}],"id":[{"id":"10.13039\/501100007514","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007514","name":"Universit\u00c3 di Pisa","doi-asserted-by":"publisher","award":["PRA_2018_5 Geometric"],"award-info":[{"award-number":["PRA_2018_5 Geometric"]}],"id":[{"id":"10.13039\/501100007514","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007514","name":"Universit\u00c3 di Pisa","doi-asserted-by":"publisher","award":["algebraic and analytic methods in arithmetic (unipi)"],"award-info":[{"award-number":["algebraic and analytic methods in arithmetic (unipi)"]}],"id":[{"id":"10.13039\/501100007514","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The problem of finding the density of odd integers which can be expressed as the sum of a prime and a power of two is a classical one. In this paper we tackle the problem both with a direct approach and with a theoretical approach, suggested by Bombieri. These approaches were already introduced by Romani in [Calcolo 20 (1983), no. 3, pp. 319\u2013336], but here the methods are extended and enriched with statistical and numerical methodologies. Moreover, we give a proof, under standard heuristic hypotheses, of the formulas claimed by Bombieri, on which the theoretical approach is based. The different techniques produce estimates of the densities which coincide up to the first three digits.<\/p>","DOI":"10.1090\/mcom\/3537","type":"journal-article","created":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T09:39:16Z","timestamp":1583314756000},"page":"2365-2386","source":"Crossref","is-referenced-by-count":5,"title":["On computing the density of integers of the form 2\u207f+\ud835\udc5d"],"prefix":"10.1090","volume":"89","author":[{"given":"Gianna","family":"Del Corso","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ilaria","family":"Del Corso","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roberto","family":"Dvornicich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francesco","family":"Romani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2020,4,28]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"363","DOI":"10.2307\/2004056","article-title":"A heuristic asymptotic formula concerning the distribution of prime numbers","volume":"16","author":"Bateman, Paul T.","year":"1962","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"P. T. Bateman and R. A. Horn, Primes represented by irreducible polynomials in one variable. Proc. Symp, in Pure Math. 8 (1965), 119\u2013132.","DOI":"10.1090\/pspum\/008\/0176966"},{"key":"3","unstructured":"E. Bombieri, Private communication, 1977."},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"242","DOI":"10.2307\/2683386","article-title":"Algorithms for computing the sample variance: analysis and recommendations","volume":"37","author":"Chan, Tony F.","year":"1983","journal-title":"Amer. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-1305","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.jnt.2003.11.009","article-title":"On Romanoff\u2019s constant","volume":"106","author":"Chen, Yong-Gao","year":"2004","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"6","unstructured":"A. de Polignac, R\u00e9ch\u00e8rches nouvelles sur les nombres premieres, C. R. Acad. Sci. Paris S\u00e9r. 29 (1849), \"397\u2013401\"."},{"key":"7","first-page":"113","article-title":"On integers of the form 2^{\ud835\udc58}+\ud835\udc5d and some related problems","volume":"2","author":"Erd\u00f6s, P.","year":"1950","journal-title":"Summa Brasil. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-498X","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"45","DOI":"10.4064\/aa122-1-4","article-title":"On integers of the form \ud835\udc5d+2^{\ud835\udc58}","volume":"122","author":"Habsieger, Laurent","year":"2006","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027","volume-title":"Accuracy and stability of numerical algorithms","author":"Higham, Nicholas J.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0898715210","edition":"2"},{"key":"10","doi-asserted-by":"crossref","unstructured":"W. Kahan, Further remarks on reducing truncation errors, Commun. ACM 8 (1965), no. 1, 40.","DOI":"10.1145\/363707.363723"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"137","DOI":"10.4064\/aa135-2-3","article-title":"The Romanoff theorem revisited","volume":"135","author":"Li, Hongze","year":"2008","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"12","unstructured":"M. Madritsch and S. Planitzer, Romanov\u2019s theorem in number fields, Tech. report,  arXiv:arXiv:1512.04869, 2015."},{"issue":"275","key":"13","doi-asserted-by":"publisher","first-page":"1849","DOI":"10.1090\/S0025-5718-2011-02445-2","article-title":"On the integers of the form \ud835\udc5d\u00b2+\ud835\udc4f\u00b2+2\u207f and \ud835\udc4f\u00b2\u2081+\ud835\udc4f\u00b2\u2082+2^{\ud835\udc5b\u00b2}","volume":"80","author":"Pan, Hao","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-2","key":"14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10474-006-0060-6","article-title":"A note on Romanov\u2019s constant","volume":"112","author":"Pintz, J.","year":"2006","journal-title":"Acta Math. Hungar.","ISSN":"https:\/\/id.crossref.org\/issn\/0236-5294","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/BF02576468","article-title":"Computations concerning primes and powers of two","volume":"20","author":"Romani, F.","year":"1983","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"key":"16","unstructured":"F. Romani, Computer techniques applied to the study of additive sequences, Ph.D. thesis, Scuola Normale Superiore di Pisa, 1978. Supervisor: E. Bombieri."},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"668","DOI":"10.1007\/BF01449161","article-title":"\u00dcber einige S\u00e4tze der additiven Zahlentheorie","volume":"109","author":"Romanoff, N. P.","year":"1934","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4064\/aa-7-1-1-8","article-title":"Remarks on the paper \u201cSur certaines hypoth\u00e8ses concernant les nombres premiers\u201d","volume":"7","author":"Schinzel, A.","year":"1961","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"445","DOI":"10.2307\/2004008","article-title":"A remark on a paper of Bateman and Horn","volume":"17","author":"Schinzel, A.","year":"1963","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"20","first-page":"99","article-title":"On de Polignac\u2019s conjecture","volume":"27","author":"van der Corput, J. G.","year":"1950","journal-title":"Simon Stevin","ISSN":"https:\/\/id.crossref.org\/issn\/0037-5454","issn-type":"print"},{"key":"21","series-title":"Cambridge Tracts in Mathematics","isbn-type":"print","volume-title":"The Hardy-Littlewood method","volume":"80","author":"Vaughan, R. C.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0521234395"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2020-89-325\/S0025-5718-2020-03537-6\/mcom3537_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/earlyview\/#mcom3537\/.pdf","content-type":"unspecified","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2020-89-325\/S0025-5718-2020-03537-6\/S0025-5718-2020-03537-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:46:44Z","timestamp":1776829604000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2020-89-325\/S0025-5718-2020-03537-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,28]]},"references-count":21,"journal-issue":{"issue":"325","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["S0025-5718-2020-03537-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3537","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2020,4,28]]}}}