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They conjectured that for each\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$k\\ge 2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , if one changes\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$o(X^{1\/k})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\/<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    elements of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$M_k'=\\{x^k+1: x \\in \\mathbb {N}\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>M<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msubsup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:msup>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>:<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>\u2208<\/mml:mo>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    up to\n                    <jats:italic>X<\/jats:italic>\n                    , then the resulting set cannot be written as a product set\n                    <jats:italic>AB<\/jats:italic>\n                    nontrivially. In this paper, we confirm a more general version of their conjecture for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$k\\ge 3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s00493-025-00193-9","type":"journal-article","created":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T11:32:34Z","timestamp":1767785554000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Multiplicative irreducibility of small perturbations of the set of shifted k-th powers"],"prefix":"10.1007","volume":"46","author":[{"given":"Chi Hoi","family":"Yip","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,7]]},"reference":[{"key":"193_CR1","doi-asserted-by":"crossref","unstructured":"Batta, G.o., Hajdu, L., Pongr\u00e1cz, A.: On Diophantine graphs. 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