Solutions of the Diophantine Equation Jr = Ms +Mt

We determine all nonnegative integer solutions ( r , s , t ) , with s > t , of the Diophantine equation J r = M s + M t , where M k and J k denote the Millatt and Jacobsthal sequences, respectively. Explicit Binet formulas turn the equation into two nonzero linear forms in logarithms of algebraic numbers. Applying Matveev’s lower bound first produces an effective, although large, upper bound on the indices. We then use the Dujella–Pethő continued-fraction reduction method in two stages: first to bound the difference s − t , and subsequently to bound t . These estimates reduce the problem to a finite range that is checked by exact integer arithmetic, with independently reproducible computations. We obtain precisely three solutions: ( r , s , t ) = ( 4 , 1 , 0 ) , ( 5 , 3 , 2 ) , and ( 6 , 5 , 0 ) .