Testwiki:Reference desk/Archives/Mathematics/2008 November 28

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November 28

Capital pi notation

Hi. What means :

$\frac{F_n(x,c)}{\prod _m{G}_m(x,c)},n:m$

I do not know how to read Pi notation here.

it is from paper Andrey Morozov : Universal Mandelbrot Set as a Model of Phase Transition Theory--Adam majewski (talk) 13:20, 28 November 2008 (UTC)

The denominator is the product of the G_m's. See Multiplication#Capital pi notation. Zain Ebrahim (talk) 13:25, 28 November 2008 (UTC)

You've just inspired me to eat a big piece of pie. :-) StuRat (talk) 15:54, 28 November 2008 (UTC)

Maybe the whole formule will show the problem better ( it is a product of divisors of m or somthing like that ) :

$G_n(x,c)=\frac{F_n(x,c)}{\prod _m{G}_m(x,c)},n:m$

This is not explained on the page about multiplication. --Adam majewski (talk) 16:19, 28 November 2008 (UTC)

Sorry, I thought you just wanted to know what the capital pi means. I have no idea what those three vertical dots are for - maybe they mean n divides m (as though it was a vertical bar) so the m's in the denominator are all the multiples of n. Just guessing. Zain Ebrahim (talk) 16:42, 28 November 2008 (UTC)

The first section of the paper you linked to is saying the following. We define:

$f(x,c)=x^2+c$

$F_n(x,c)=f^n(x,c)-x$

so that the set of solutions to F_n(x,c)=0 for a given value of c are the fixed points of fⁿ(x,c). Some of these points will have order n under the iterated map x→f(x,c) - but some of them will have an order m that is a factor of n. So we define G_n(x,c) to be the polynomial whose roots are the points with order exactly n. Then

$\prod _{m|n}{G}_m(x,c)=F_n(x,c)$

because every root of F_n(x,c) is a root of some G_m(x,c) for some m that is a factor of n - and possibly m=n. So

$G_n(x,c)=\frac{F_n(x,c)}{\prod _{m|n,m<n}{G}_m(x,c)}$

You can use a similar process to iteratively find the cyclotomic polynomials. n:m seems to mean m is a proper divisor of n. Gandalf61 (talk) 17:39, 28 November 2008 (UTC)

You can give an explicit (non-recursive) expression for G in terms of F using the Möbius inversion formula: $G_n(x,c)=\prod _{d|n}{F}_{\frac{n}{d}}(x,c{)}^{\mu (d)}$, where $\mu$ is the Möbius function. Algebraist 17:48, 28 November 2008 (UTC)

Thx for answers. (:-)) See also :

• [Do1] V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?
• [Do2] V Dolotin , A Morozow : Algebraic Geometry of Discrete Dynamics. The case of one variable

--Adam majewski (talk) 08:18, 29 November 2008 (UTC)

Name of theorem/conjecture

What is the name of the conjecture/theorem that says:

For any positive integer k, there are only a finite number of positive integer solutions (a, b, c, d) such that $a^b-c^d=k$.

Dragons flight (talk) 19:19, 28 November 2008 (UTC)

According to Catalan's conjecture, it is Pillai's conjecture. JackSchmidt (talk) 20:01, 28 November 2008 (UTC)

Thanks. Dragons flight (talk) 20:26, 28 November 2008 (UTC)

Google Billboard Riddle

Several years ago, Google ran a billboard ad looking for potential employees:

[1] (work-safe)

The question asked for the "first 10-digit prime found in consecutive digits of e".

How does one solve this?

Acceptable (talk) 22:43, 28 November 2008 (UTC)

You write a computer program to go looking for it, either equipped with the decimal expansion of e and a list of all 10-digit primes or with algorithms for generating digits of e and testing 10-digit numbers for primality. There's nothing you can do smarter than just searching that I'm aware of. Algebraist 22:48, 28 November 2008 (UTC)

The prime number theorem says that about 4% of ten-digit numbers can be expected to be prime, so you probably won't have to go very far in e. It's not worth having a list of prime ten-digit numbers. For each ten-digit number a you want to test, divide by each prime p between 2 and 100,000 (the square root of 10¹⁰). The first a you find that's not divisible by any of these is your number. 67.150.252.236 (talk) 23:01, 28 November 2008 (UTC)

There are much better primality tests if you need speed. I'm not sure if that'll matter hear, though, since it's a fairly small problem. Algebraist 23:25, 28 November 2008 (UTC)

Or the lazy man's way: interpret the question as the "first ten digit prime, that is found in consecutive digits of e" first meaning first in the usual order of natural numbers, then lazily assume e is a Normal number and just give the smallest ten digit prime, whatever it is. This would probably pick out employees who would either save them a lot of money, or embezzle a lot. :-)John Z (talk) 12:17, 29 November 2008 (UTC)

The number 0000000002 is definitely prime, and the concept of a 10-digit may be considered context-dependent. Bo Jacoby (talk) 23:21, 29 November 2008 (UTC).

Actually harder than one might think. I found 7427466391 which starts 98 digits in. Dragons flight (talk) 23:35, 29 November 2008 (UTC)

The easier way to solve it is to look at our article, which says that "a random stream of digits has a 98.4% chance of starting a 10-digit prime sooner." Yes, Wikipedia does, in fact, contain all of the information in the universe. « Aaron Rotenberg « Talk « 03:40, 30 November 2008 (UTC)

I put "first 10-digit prime found in consecutive digits of e" into the little search box I have for google on my browser and looked at the first answer that came up and it gave the answer. Google contains all the information in the universe and a lot besides :) I'm sure that must be easier. Dmcq (talk) 23:30, 30 November 2008 (UTC)