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diophantine eqn

Source: Miklos Schweitzer 2005 q3

August 10, 2021
number theory

Problem Statement

Let α≤22\alpha\leq 22 be non-negative integer. For which α\alpha does the equation 8x23−5αy23=18x^{23}-5^{\alpha}y^{23}=1 have the most integer solutions (x,y)? What can we say about α≥23\alpha\geq 23?
I believe the eqn has solutions only when α=0\alpha=0. taking modulo 47, α≡9,17(mod23)\alpha\equiv 9,17\pmod{23} or (23∣α23|\alpha and 47∣x47|x). taking modulo 139 and 277 eliminates the α≡9,17(mod23)\alpha\equiv 9,17\pmod{23} cases. 139=23*6+1 , 277=23*12+1
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