Type: Article
Publication Date: 2021-08-27
Citations: 3
DOI: https://doi.org/10.1515/crelle-2021-0048
Abstract It is proved that all sufficiently large integers n can be represented as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>13</m:mn> <m:mn>14</m:mn> </m:msubsup> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> n=x_{1}^{2}+x_{2}^{3}+\cdots+x_{13}^{14}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>13</m:mn> </m:msub> </m:mrow> </m:math> {x_{1},\ldots,x_{13}} are positive integers. This improves upon the current record with fourteen variables in place of thirteen.