Determine all possible pairs (A,B) of positive integers that satisfy this equation:
Provide valid reasoning for your answer.
Notes:
(i) x^x^x is equal to x^(x^x) and NOT (x^x)^x.
(ii) Computer program/excel solver assisted solutions are welcome, but a semi-analytic (hand calculator and p&p) methodology is preferred.
(iii) Adapted from a problem appearing in Taiwan M.O: 2000.
If A is 1, RHS is always 1, so B can only be 1
(1,1) is a solution
If A is 2, RHS is 2^(B+2), LHS is B^4. B=2 is the only integer solution
(2,2) is a solution
If A is 3, RHS is 3^(B+2) = 9*3^B
LHS is B^9. No integer solutions I can find.
Seems unlikely there are any other solutions.
So, I'm finding only {(1,1), (2,2)}, but I have not proved there are no others.
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Posted by Larry
on 2023-08-18 11:30:53 |
Analytic solution; not proved complete
