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\frac{2016^m-m^{2016}}{m+2016} is integer

Source: 2016 Thailand October Camp 3.2

February 28, 2022
number theoryDivisibility

Problem Statement

(i)\text{(i)} Does there exist a positive integer m>20162016m > 2016^{2016} such that 2016m−m2016m+2016\frac{2016^m-m^{2016}}{m+2016} is a positive integer?
(ii)\text{(ii)} Does there exist a positive integer m>20172017m > 2017^{2017} such that 2017m−m2017m+2017\frac{2017^m-m^{2017}}{m+2017} is a positive integer?
(Serbia MO 2016 P1)
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