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I am preparing for job interviews, and in some old questions list I found this interesting one which I did not manage to crack yet:

Compute the following products: $$ P_1 = \Pi_{0< i<j<\infty} (i^{1/i} - j^{1/j}) $$ $$ P_2 = \Pi_{0< i<j<2020} (i^{1/i} - j^{1/j}) $$

Focussing of $P_1$, I can easily analyze the asymptotic behaviour for which both $i^{1/i} = j^{1/j} \rightarrow 1$ and thus $i^{1/i} - j^{1/j} \rightarrow 0$, indicating that it should be $P_1 =0$. However computing $P_2$ seems very complicated. I am sure that there must be some kind of trick to trivialize the product that I am still missing. Could anybody help?

Solution: given the hint in the answer by Lorago, it is immediate to see that the product contains a factor of zero and hence $P_1 = P_2 = 0$.

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    $\begingroup$ Hint: there's a zero in the factors, so everything does trivialize. $\endgroup$
    – Trebor
    Apr 15 at 9:04
  • $\begingroup$ Bonus exercise: you now likely know how to solve $x^y = y^x$ for positive integers $x,y$. $\endgroup$
    – chi
    Apr 15 at 17:07

1 Answer 1

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Hint:

$$4^{1/4}=(4^{1/2})^{1/2}=2^{1/2}$$

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