What would be some exponential Diophantine equations for the beginner to solve (which can demonstrate the techniques?) especially good if there are hints! Thank you very much!
Exponential Diophantine Equations for Beginners
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number-theory
diophantine-equations
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0There is$a$very interesting tool to solve this type of problem, the groups chosen. – 2017-07-23
1 Answers
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The posed problem is tightly connected with FLT, which here is not examined. But it is it's a pity! However,… If Fermat’s equality exists, then in the numeration system with the prime base n>2 next-to-last digits in numbers $1^n$, $2^n$,...$(n-1)^n$ are equal to 0 and, therefore, the two-digit end of the number $S=1^n+2^n+...+(n-1)^n$ is equal to the sum of the arithmetical progression S'=1+2+...+(n-1), i.e. is equal to the number $d0$, where the digit $d$ is not zero. That contradicts the direct calculation of the end of the number S (it is equal to 00, which is evident when grouping the terms of the sum $S$ into the pairs: $S=[1^n+(n-1)^n]+[2^n+(n-2)^n]+...)$.