Commutators and associators of positive integer exponents

Asked Oct 31 '17 at 13:59

Active Oct 31 '17 at 14:14

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There is a pair $(2, 4)$ for which $2 ^ 4 = 4 ^ 2$. I wonder if there is another such pair $$(n, m): n ^ m = m ^ n$$ It almost obvious that is not. But I failed to prove it.

The question is: does one studied a commutator and associator of ^ operation? Where can I read about it? Is there another such pair? pairs?

edited Oct 31 '17 at 14:14

Peter

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asked Oct 31 '17 at 13:59

vlad9486

1

Hint: look at the function $f(x)=\frac {\log x}x$. – lulu Oct 31 '17 at 14:03
If positive real numbers are allowed, we have infinite many non-trivial solutions, but for positive integers , the only non-trivial pair is $(2/4)$ – Peter Oct 31 '17 at 14:13
@lulu Is it known whether there is another non-trivial solution in positive rational numbers ? – Peter Oct 31 '17 at 14:16
Looks like even on real numbers no non-trivial solutions. – vlad9486 Oct 31 '17 at 14:24
m ^ n = exp(n * ln(m)) n ^ m = exp(m * ln(n)) so m ^ n = n ^ m => n * ln(m) = m * ln(n) => f(m) = f(n), where f(x) = ln(x) / x – vlad9486 Oct 31 '17 at 14:25
but... f(x) is monotonic – vlad9486 Oct 31 '17 at 14:26
@vlad9486 Increasing for $x\le e$ and decreasing for $x\ge e$. Consider for example the pair $(1.7649219\cdots /5)$ – Peter Oct 31 '17 at 14:29
Yes, you are right, the first implication is incorrect. – vlad9486 Oct 31 '17 at 14:40
See also this question. – Steve D Oct 31 '17 at 16:00

Commutators and associators of positive integer exponents

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