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A comment on Tell HN: 2^4 == 4^2, extended to rationals.

Re:Tell HN: 2^4 == 4^2, extended to rationals by nmmnthrowaway on Saturday April 22, 2023 @02:12PM (#35669402)
I think a more interesting problem to consider after solving n^m = m^n is solving n^m = m^n + 1 in the natural numbers.
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Re:Tell HN: 2^4 == 4^2, extended to rationals by mahalex on Saturday April 22, 2023 @02:26PM (#35669519)
If we exclude 0, the only solutions are 2^1 = 1^2 + 1 and 3^2 = 2^3 + 1
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Re:Tell HN: 2^4 == 4^2, extended to rationals by nmmnthrowaway on Saturday April 22, 2023 @02:30PM (#35669550)
That is correct. Why are there no others?
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Re:Tell HN: 2^4 == 4^2, extended to rationals by mahalex on Saturday April 22, 2023 @02:36PM (#35669615)
Because Catalan’s conjecture.
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Re:Tell HN: 2^4 == 4^2, extended to rationals by yarg on Saturday April 22, 2023 @02:53PM (#35669772)
> Catalan's conjecture was proven by Preda Mihăilescu in April 2002.

Cool, so the conjecture's a theorem.

https://xn--uni-gttingen-8ib.academia.edu/PredaMihailescu

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Re:Tell HN: 2^4 == 4^2, extended to rationals by hgsgm on Saturday April 22, 2023 @02:57PM (#35669815)
Which has an extremely complicated proof :-(
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Re:Tell HN: 2^4 == 4^2, extended to rationals by mahalex on Saturday April 22, 2023 @04:07PM (#35670433)
I wouldn’t call it extremely complicated; it is much simpler than the proof of FLT.
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Re:Tell HN: 2^4 == 4^2, extended to rationals by nmmnthrowaway on Saturday April 22, 2023 @03:06PM (#35669909)
Yes, but there is a much simpler proof.
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Re:Tell HN: 2^4 == 4^2, extended to rationals by judofyr on Saturday April 22, 2023 @03:21PM (#35670070)
Something related to prime factorization maybe?
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Re:Tell HN: 2^4 == 4^2, extended to rationals by AnimalMuppet on Saturday April 22, 2023 @03:29PM (#35670149)
Do tell.
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