{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:23:29Z","timestamp":1776864209146,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,2,25]],"date-time":"2012-02-25T00:00:00Z","timestamp":1330128000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove that the sumset\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace p squared plus b squared plus 2 Superscript n Baseline colon p is prime and b comma n element-of double-struck upper N right-brace\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mtext>\u00a0is prime and\u00a0<\/mml:mtext>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{p^2+b^2+2^n:\\, p\\text { is prime and } b,n\\in \\mathbb {N}\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has positive lower density. We also construct a residue class with an odd modulus that contains no integer of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p squared plus b squared plus 2 Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p^2+b^2+2^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Similar results are established for the sumset\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace b 1 squared plus b 2 squared plus 2 Superscript n squared Baseline colon b 1 comma b 2 comma n element-of double-struck upper N right-brace period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{b_1^2+b_2^2+2^{n^2}:\\, b_1,b_2,n\\in \\mathbb {N}\\}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02445-2","type":"journal-article","created":{"date-parts":[[2011,2,25]],"date-time":"2011-02-25T09:49:20Z","timestamp":1298627360000},"page":"1849-1864","source":"Crossref","is-referenced-by-count":2,"title":["On the integers of the form \ud835\udc5d\u00b2+\ud835\udc4f\u00b2+2\u207f and \ud835\udc4f\u2081\u00b2+\ud835\udc4f\u2082\u00b2+2^{\ud835\udc5b\u00b2}"],"prefix":"10.1090","volume":"80","author":[{"given":"Hao","family":"Pan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,2,25]]},"reference":[{"key":"1","unstructured":"J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of \ud835\udc4f\u207f\u00b11,\ud835\udc4f=2,3,5,6,7,10,11,12 up to High Powers, 3rd ed., Contemporary Mathematics 22, Amer. Math, Soc., Providence, RI, 2002."},{"key":"2","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1515\/crll.1994.454.59","article-title":"Lagrange\u2019s four squares theorem with almost prime variables","volume":"454","author":"Br\u00fcdern, J\u00f6rg","year":"1994","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1613","DOI":"10.1090\/S0002-9939-99-05482-9","article-title":"On integers of the form 2^{\ud835\udc58}\u00b1\ud835\udc5d^{\ud835\udefc\u2081}\u2081\ud835\udc5d^{\ud835\udefc\u2082}\u2082\u22ef\ud835\udc5d^{\ud835\udefc\u1d63}\u1d63","volume":"128","author":"Chen, Yong-Gao","year":"2000","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1090\/S0002-9939-00-05916-5","article-title":"On integers of the form \ud835\udc582\u207f+1","volume":"129","author":"Chen, Yong-Gao","year":"2001","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1006\/jnth.2001.2640","article-title":"On integers of the forms \ud835\udc58-2\u207f and \ud835\udc582\u207f+1","volume":"89","author":"Chen, Yong-Gao","year":"2001","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1016\/S0022-314X(02)00051-3","article-title":"On integers of the forms \ud835\udc58^{\ud835\udc5f}-2\u207f and \ud835\udc58^{\ud835\udc5f}2\u207f+1","volume":"98","author":"Chen, Yong-Gao","year":"2003","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"250","key":"7","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1090\/S0025-5718-04-01674-6","article-title":"Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers","volume":"74","author":"Chen, Yong-Gao","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.jnt.2006.10.005","article-title":"On integers of the forms \ud835\udc58\u00b12\u207f and \ud835\udc582\u207f\u00b11","volume":"125","author":"Chen, Yong-Gao","year":"2007","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"51","DOI":"10.4064\/aa135-1-4","article-title":"Fermat numbers and integers of the form \ud835\udc4e^{\ud835\udc58}+\ud835\udc4e^{\ud835\udc59}+\ud835\udc5d^{\ud835\udefc}","volume":"135","author":"Chen, Yong-Gao","year":"2008","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"79","DOI":"10.2307\/2005463","article-title":"Not every number is the sum or difference of two prime powers","volume":"29","author":"Cohen, Fred","year":"1975","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","first-page":"103","DOI":"10.2140\/pjm.1971.36.103","article-title":"On the sum of a prime and of two powers of two","volume":"36","author":"Crocker, Roger","year":"1971","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"235","DOI":"10.4064\/cm112-2-3","article-title":"On the sum of two squares and two powers of \ud835\udc58","volume":"112","author":"Crocker, Roger Clement","year":"2008","journal-title":"Colloq. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-1354","issn-type":"print"},{"key":"13","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":"14","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"},{"key":"15","doi-asserted-by":"publisher","first-page":"203","DOI":"10.4064\/aa-21-1-203-234","article-title":"Primes of the type \ud835\udf19(\ud835\udc65,\ud835\udc66)+\ud835\udc34 where \ud835\udf19 is a quadratic form","volume":"21","author":"Iwaniec, H.","year":"1972","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"69","DOI":"10.4064\/aa-29-1-69-95","article-title":"The half dimensional sieve","volume":"29","author":"Iwaniec, H.","year":"1976","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"171","DOI":"10.4064\/aa-36-2-171-202","article-title":"Rosser\u2019s sieve","volume":"36","author":"Iwaniec, Henryk","year":"1980","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"18","doi-asserted-by":"publisher","first-page":"307","DOI":"10.4064\/aa-37-1-307-320","article-title":"A new form of the error term in the linear sieve","volume":"37","author":"Iwaniec, Henryk","year":"1980","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"19","series-title":"London Mathematical Society Monographs, No. 4","volume-title":"Sieve methods","author":"Halberstam, H.","year":"1974"},{"key":"20","unstructured":"E. Landau, \u00dcber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math. Phys., 13(1908), 305-312."},{"issue":"7","key":"21","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1090\/S0002-9939-05-07827-5","article-title":"Fibonacci numbers that are not sums of two prime powers","volume":"133","author":"Luca, Florian","year":"2005","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s006050050065","article-title":"Squares of primes and powers of 2","volume":"128","author":"Liu, Jianya","year":"1999","journal-title":"Monatsh. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-9255","issn-type":"print"},{"key":"23","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3845-2","volume-title":"Additive number theory","volume":"164","author":"Nathanson, Melvyn B.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/038794656X"},{"key":"24","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1515\/crll.1968.231.89","article-title":"\u00dcber die Summe aus einem Quadrat und einem Primzahlquadrat","volume":"231","author":"Rieger, G. J.","year":"1968","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"1","key":"25","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"},{"issue":"4","key":"26","doi-asserted-by":"publisher","first-page":"997","DOI":"10.1090\/S0002-9939-99-05502-1","article-title":"On integers not of the form \u00b1\ud835\udc5d^{\ud835\udc4e}\u00b1\ud835\udc5e^{\ud835\udc4f}","volume":"128","author":"Sun, Zhi-Wei","year":"2000","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"183","DOI":"10.4064\/aa99-2-5","article-title":"Integers not of the form \ud835\udc50(2^{\ud835\udc4e}+2^{\ud835\udc4f})+\ud835\udc5d^{\ud835\udefc}","volume":"99","author":"Sun, Zhi-Wei","year":"2001","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1017\/S0013091500000924","article-title":"A note on integers of the form 2\u207f+\ud835\udc50\ud835\udc5d","volume":"45","author":"Sun, Zhi-Wei","year":"2002","journal-title":"Proc. Edinb. Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-0915","issn-type":"print"},{"issue":"267","key":"29","doi-asserted-by":"publisher","first-page":"1853","DOI":"10.1090\/S0025-5718-09-02212-1","article-title":"Covers of the integers with odd moduli and their applications to the forms \ud835\udc65^{\ud835\udc5a}-2\u207f and \ud835\udc65\u00b2-\ud835\udc39_{3\ud835\udc5b}\/2","volume":"78","author":"Wu, Ke-Jian","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"23","DOI":"10.4064\/aa115-1-2","article-title":"Integers not of the form \ud835\udc50(2^{\ud835\udc4e}+2^{\ud835\udc4f})+\ud835\udc5d^{\ud835\udefc}","volume":"115","author":"Yuan, Pingzhi","year":"2004","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02445-2\/S0025-5718-2011-02445-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02445-2\/S0025-5718-2011-02445-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:53:37Z","timestamp":1776790417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02445-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,25]]},"references-count":30,"journal-issue":{"issue":"275","published-print":{"date-parts":[[2011,7]]}},"alternative-id":["S0025-5718-2011-02445-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02445-2","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":[[2011,2,25]]}}}