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The number of solutions to the generalized Pillai equation ± r a x ± s b y = c .

Reese ScottRobert Styer — 2013

Journal de Théorie des Nombres de Bordeaux

We consider N , the number of solutions ( x , y , u , v ) to the equation ( - 1 ) u r a x + ( - 1 ) v s b y = c in nonnegative integers x , y and integers u , v { 0 , 1 } , for given integers a > 1 , b > 1 , c > 0 , r > 0 and s > 0 . When gcd ( r a , s b ) = 1 , we show that N 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) < 2 · 10 15 for each solution; when gcd ( a , b ) > 1 , we show that N 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N = 3 solutions.

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