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Monday, March 5, 2018

Problem of the Week 3-6-18: Diophantine Equation/Prime Powers [Number Theory]

Check out this Problem of the Week.
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Solution below.




  1. Hi. I understand the part on factorising p out of x^n and y^n till one of them cannot be divided p anymore. However, why does it necessarily imply that n=pt? So far we have yet to discuss anything related to the power. Do you use something like Fermat Little Theorem?

    1. After factoring out p from x^n and y^n until p does not divide one of them anymore, we also note that they must both have the same number of factors of p, otherwise they would not satisfy the equation. We can use this as well as our conclusion that nx^(n-1) is congruent to 0 mod p, to then conclude that n must be a multiple of p, since x cannot be a multiple of p (nx^(n-1) congruent to 0 mod p is equivalent to the statement that p divides nx^(n-1)).