Use modular arithmetic to solve an exponential Diophantine equation: 7^x=3*2^y+1

This video introduces the concept of modular arithmetic as a tool to solve Diophantine equations that contain exponent functions: 7^x=3·2^y+1. If shows how to choose a proper modulus in a cogruence expression and how to simplify the congruence expresions to lead to useful conclusions, which will be used in another congruence expression until a final conclusion is made.

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if you've already excluded the case x being odd, you can put 1 to the other side, writing x = 2*x0, split the difference of squares, as (7^x0 - 1)(7^x0 + 1). these 2 terms are even and apart by 2, so must the right hand side 3*2^y. the only possible case is 3*2^1 and 2^3 (6 and 8)