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June 17

What does this "mean" ?

The first integral below becomes the expression of Euler's famous gamma function when making the simple substitution ${\sqrt[{k}]{x}}\ \to \ t.$ . The second one has already been studied by John Wallis three or four centuries ago (see here, on page 49), and its expression becomes that of Euler's well-known beta function by making the exact same substitution.

\int _{0}^{\infty }{e^{-{\sqrt[{k}]{x}}}}\ dx\ =\ k!\ ,\qquad \qquad \int _{0}^{1}{\left(1-{\sqrt[{k}]{x}}\right)^{n-k}}\ dx\ =\ {\frac {1}{C_{n}^{k}}}\ ,\qquad \qquad e^{-{\sqrt[{k}]{x}}}\ {\xrightarrow[{\text{Taylor series}}]{\quad {\sqrt[{k}]{x}}\ \to \ 0\quad }}\ 1-{\sqrt[{k}]{x}}\ .

I somehow have the nagging feeling that "there must something to this", but I can't quite figure out just what "it" is, exactly... Like not being able to see the forest for the trees... Could somebody help me in connecting the dots ? — 79.113.242.1 (talk) 18:54, 17 June 2013 (UTC)[reply]

Unresolved

Computational limits

After looking at Fermat-Catalan conjecture, and noting that the highest known solution only has $c^{k}\approx 9\times 10^{14}$ , I decided that I could burn some idle CPU cycles and test that. So I wrote some brute force code and let it run for a while to establish there are no additional solutions for $c^{k}<10^{20}$ . Somehow, I suspect this isn't the best limit known to man, but there is no discussion of computational searches in the article, and so far my Google Fu has been unable to identify any specific limit on searches for Fermat-Catalan solutions. Can anyone find such a limit? More generally, are there places where people publish negative results of computational searches in number theory? Or are such results just considered too boring to bother with? Dragons flight (talk) 23:49, 17 June 2013 (UTC)[reply]

Regarding negative results of computational searches in number theory, they are definitely not too boring to bother with (see for example Wieferich prime#History and search status and the references for this section). Also, some people spend many CPU hours on trying to expand some computationally hard sequences in the On-Line Encyclopedia of Integer Sequences. Some of these searches might not yield any positive results for a long time, but they prove that no positive solution exists in the search area. -- Toshio Yamaguchi 15:26, 20 June 2013 (UTC)[reply]