Wikipedia:Reference desk/Archives/Mathematics/2013 September 24

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September 24

Diagram

What you can or cannot conclude from a diagram. — Preceding unsigned comment added by Mike Boissonnault (talk • contribs) 00:50, 24 September 2013 (UTC)

That would rather depend on the diagram, wouldn't it ? Care to give us an example ? StuRat (talk) 13:09, 24 September 2013 (UTC)

In formal mathematics, a diagram may be used to illustrate a proof, but no conclusions should be drawn from the diagram as it may not be completely general or sufficiently accurate. For example, you cannot draw a completely general triangle, as the specific triangle in your diagram will be either a right triangle, an acute triangle or an obtuse triangle - it cannot be all three at the same time. The missing square puzzle shows the dangers of relying on diagrams - the puzzle relies on an apparently straight line which is in fact not quite straight. In less formal contexts, proofs without words and diagrammatic notations such as Feynmann diagrams, Penrose diagrams and Penrose graphical notation can all be useful. Gandalf61 (talk) 14:14, 24 September 2013 (UTC)

• In case you are talking in the category sense, our article Diagram (category theory) would be a start; but the question would be hard to answer without a little more specificity.
• In the generic sense of geometric diagrams/pictures, I would say that you shouldn't conclude anything from them, but that they can be useful for inspiration/insight. In the same way that working out specific example cases can be of value, but does not constitute proof- and, sometimes, a diagram can be misleading, so you should be cautious on over using them (sort of like how observing that 3, 5, 7, 11, and 13 are prime might make you think that all nonsquare odd numbers are prime.)
• A more formal use of diagrams in logic can be found at our article Method of analytic tableaux.
• Visual programming language gives examples of programming languages that interpret flowcharts, and other such.
• There is an interactive proof system Diamond that uses diagrams to construct concrete cases, via interaction with the user, then generalizes and checks the proof. See here: [1], [2].
• Pierce did some work with this. See our articles: Logical graph and Existential graph. For more, you might consult, [3].
• For more information on reasoning and diagrams, etc. Have a look at: Diagrammatic reasoning, Mathematical diagram, [4], [5], and [6]. (the attached pdfs were free and online, I'm sure there are many more comprehensive sources elsewhere if you have an interest).
• You may also want to take a look at Graph Theory (a branch of mathematics whose object of study is related to how connections can be modeled with a diagram- that's a bit glib, though). And, in a more general context, Mathematical visualization. Phoenixia1177 (talk) 05:24, 27 September 2013 (UTC)

True or False ?

Would anyone be so kind as to verify the following equality, either confirming it or infirming it ?

$\int _{0}^{\infty }{\frac {f(x)}{F(x)}}\ dx\ =\ \int _{0}^{\infty }{\frac {x!}{x^{x}}}\ dx$

where

$f(x)\ =\ \sum _{n=1}^{\infty }{\frac {n^{x}}{n^{n}}}\qquad ,\qquad F(x)\ =\ \sum _{n=1}^{\infty }{\frac {n^{x}}{n!}}$

Thank you ! — 79.113.233.194 (talk) 01:46, 24 September 2013 (UTC)

Sideways discussions
The following discussion has been closed. Please do not modify it.
The equation is meaningless as written as the factorial function only applies to non-negative integers, so can't be used on a variable of integration. You could perhaps replace $x!$ by $\Gamma (x+1)\$. AndrewWTaylor (talk) 13:22, 24 September 2013 (UTC) I think Arfken's mathematical physics text suggests the reverse: use x! as a synonym for Γ(x+1).76.218.104.120 (talk) 22:48, 24 September 2013 (UTC) $n!\ =\ \int _{0}^{\infty }e^{-{\sqrt[{n}]{x}}}dx\qquad ,\qquad \forall \ n\in \mathbb {R} _{+}\quad -$ Satisfied ? — 79.113.229.58 (talk) 13:41, 24 September 2013 (UTC) Does everyone else see a red bold message about "Failed to parse (unknown error)" above (in two places)? (I don't know how it could be just me but no one else is commenting about it or fixing it...) RJFJR (talk) 13:44, 24 September 2013 (UTC) I had the same problem when I was writing my comment - I seemed to get the errors every other time I saved.. AndrewWTaylor (talk) 13:48, 24 September 2013 (UTC) I've been having this problem for several weeks now, I think... — 79.113.229.58 (talk) 13:57, 24 September 2013 (UTC)

Unresolved

 – — See link

First "block" of 100 non-primes...

I'm not looking for the first prime gap >= 100, I'm looking for the first instance where every number from k*100 to (k+1)*100-1 is composite. For example, the first "block" of 10 that is all composite is 200-209.Naraht (talk) 15:55, 24 September 2013 (UTC)

In PARI/GP:

forstep(n=0,10^7,100,if(nextprime(n)-n>100,print1(n", ")))

1671800, 2637800, 3117300, 3933600, 4640600, 4652400, 5178200, 5518700, 5837400, 5845200, 6012900, 6085000, 6333800, 6376200, 6789800, 6958700, 7129900, 7565200, 7803500, 7826900, 8027700, 8367400, 8421300, 8905200, 9549000, 9708000,

time = 516 ms.

An OEIS search shows this is oeis:A136295. PrimeHunter (talk) 16:09, 24 September 2013 (UTC)

Whoosh... Why did I expect you would be the one to answer? :) Interestingly enough, I can't find a link from this oeis link to one for "millenia", but I'm guessing it would start at a *much* higher number.Naraht (talk) 17:30, 24 September 2013 (UTC)

Based on a quick manual inspection of http://www.trnicely.net/gaps/gaplist.html#MainTable and http://www.trnicely.net/gaps/gaplist.html#SuppTable, the first case of 1000 appears to be a gap of 1122 after 31068473876462989 in this entry:

1122 CFC Be.Nyman 2001 29.55 17 31068473876462989

PrimeHunter (talk) 19:32, 24 September 2013 (UTC)

So all numbers between 31068473876463000 and 31068473876464000 are prime. Thank you!Naraht (talk) 20:05, 24 September 2013 (UTC)

You mean they are all composite. PrimeHunter (talk) 11:49, 25 September 2013 (UTC)

Yes, duh!Naraht (talk) 17:24, 27 September 2013 (UTC)