Talk:Catalan's conjecture

Given that "consecutive" is not explained in the article Integer, I think that a link to a nonexistent article Consecutive_number, which may inspire somebody to write about it, is better than a link to Integer. This is especially true in light of the fact that it's more relevant to think of the numbers as natural numbers, and there's already a link to Natural_number. Finally, if the objection is that "consecutive number" is ambiguous since we might be talking about rational numbers or real numbers, my reply is that "consecutive" makes no sense in those contexts, and indeed it's standard to interpret "consecutive number" in the context of natural numbers (or integers), unless otherwise specified. &mdash; Toby Bartels, Thursday, July 18, 2002

Name of page
So shouldn't this page be moved to "Catalan's theorem" since it's been proven? --Lowellian 04:51, May 23, 2004 (UTC)
 * At least the first sentence "Catalan's conjecture is a simple conjecture in number theory..." should be "Catalan's conjecture is a simple theorem in number theory...", shouldn't it? - ReiVaX 21:26, 28 Sep 2004 (UTC)


 * Naming isn't that simple. The name of a theorem, even if it used to be a conjecture, is largely based on societal factors.  The Smith conjecture (proven for over 20 years now), is still called the "Smith conjecture", and Fermat's Last Theorem didn't become called "Wiles' theorem".  To put it simply, there is no rule that "conjecture" gets replaced by "theorem" and there certainly is no rule that says that since Mihailescu proved the Catalan conjecture, it becomes "Mihailescu's theorem".  If indeed it is known as that, then it is for societal reasons which are murky.  In fact, I kind of doubt that so famous a conjecture as this would be renamed so quickly (if ever).  "Mihailescu's theorem" in Google Scholar brings up only a handful of hits.  I propose this page be moved back to "Catalan conjecture" unless it can be demonstrated that "Mihailescu's theorem" is the common name for this now.  --C S (Talk) 01:30, 14 May 2007 (UTC)


 * I support the above argument, and very strongly suggest that the page be moved back to "Catalan conjecture". Kope (talk) 14:41, 22 November 2007 (UTC)


 * Noticing that nobody ever made the move nor commented against it, I have made the move back and cleaned up the self-references within the article.--Doug.(talk • contribs) 03:27, 4 May 2008 (UTC)

Articles
How is Pillai's conjecture the same as Mihailescu's theorem? I will change the link after a week if no one gives me an answer. Sr13 08:29, 12 November 2006 (UTC)

though Chudnowski has claimed to prove it (Pillai's conjecture)
The unsourced parenthetical statement in this article might refer to one of the Chudnovsky brothers, but a quick reading of that article doesn't make any reference to such a claim. And googling "Chudnowski" just turns up a number of clones of this article. It seems to me that the statement "though Chudnowski has claimed to prove it" needs to be removed, unless someone can come up with a source for it.&mdash;GraemeMcRaetalk 15:44, 25 November 2009 (UTC)
 * Removed. CRGreathouse (t | c) 14:30, 12 February 2010 (UTC)

Distance Two: 26
On Feb. 11h, 2010 Alonso Del Arte stated on  that 26 is the only number between a square (25) and a cube (27), and that it would seem that 26 is the only number between two powers. He has verified this up to 10^7. --Gfis (talk) 15:15, 21 February 2010 (UTC)

Pillai's conjecture
What is meant by "d is the difference of a perfect power n"?--Thn2010 (talk) 16:58, 1 March 2012 (UTC)

Terminology
"Langevin computed a value of exp exp exp exp 730 for the bound".

What does 'exp' mean in this context? Is it e (mathematical constant}, as in e^(e^(e^(e^730)))) ? DS (talk) 20:34, 18 January 2013 (UTC)


 * Yes, "exp" is standard notation for the Exponential function and that is what is meant here. Deltahedron (talk) 07:28, 19 January 2013 (UTC)

Another conjecture
If n and n+1 are both perfect powers, then n = 8. Another conjecture is: if n is a natural number and n2+n+1 is a perfect power, then n = 18, is it true? More generally, if $$\Phi_n(b)$$ is a perfect power and n>2, b>1, then (n, b) = (3, 18), (5, 3), or (6, 19). — Preceding unsigned comment added by 101.14.116.15 (talk) 13:26, 31 March 2015 (UTC)

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Errors?
Many entries in the table appear to be erroneous. For example, 250000 (distance 47, allegedly) is not a perfect power. It has 2 and 5 as factors. I tried to check several entries unsuccessfully. — Preceding unsigned comment added by Bastiaan Zapf (talk • contribs) 15:36, 1 November 2019 (UTC)
 * Just a power of some integer, $$250000 = 500^2$$ and $$250047 = 63^3$$. Bubba73 You talkin' to me? 20:07, 1 November 2019 (UTC)