Representation by sums of unlike powers
Representation by sums of unlike powers
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"> …