POTD 2005-11

From Karlsen. Math pros please do not solve this in the channel. In $\mathbb{N}$ , the positive integers, there is a Pythagorean right triangle with sides $x, y, z$ and an inscribed circle of radius $r$ .

x 2 + y 2 = z 2 and r = 3, find all possible x, y, z

Hint: There are 3 solutions. Can you prove this? solution

Sunday, Nov 13, 2005

Find indefinite integral:

$\int 1/\ln(x) - 1/\ln(x)^2 dx\,$

Saturday, Nov 12, 2005

Conjecture of nodar

For $p > 2$ , a prime, and $k$ relatively prime to p,

$\sum_{n=0}^{p-1} \mbox{Legendre symbol}\left(\frac{n^2+k}{p}\right) = -1\,$

nerdy2 has solved it and it is fairly easy if you recall your quadratic residue introduction. solution

<nerdy2> note that this suggests another problem: suppose p is a prime, p = 1 mod 4, and k is relatively prime to p, then sum(n=0..p-1) (k - n^2/p) = -1

Friday, Nov 11, 2005

From Dedekind

Find a sequence of functions $f_n : [0, 1] \to \mathbb{R}$ such that $\lim \sup f_n = 1$ , $\lim \inf f_n = 0$ and $\int f_n \to 0$ . (the lim sup and inf are pointwise).

Tuesday, Nov. 8, 2005

Consider a random walk on the integer positions k=0,1,...,2b, such that given that the position is at k at a time slot the probability to be at position k-1 at next time slot is k/(2b+1), the probability to be at position k next time slot is 1/(2b+1), the probability to be at position k+1 next time slot is (2b-k)/(2b+1). What is the stationary probability distribution (the limiting probability distribution as time tends to infinity) for the positions?

Extra Credit What is the variance of the stationary probability distribution?

Monday, Nov. 7, 2005

Let $\mathbb{Z}_m ^*$ be the residues $(\mathrm{mod}\ m)\,$ which are relatively prime to $m\,$ . $g\in\mathbb{Z}_m ^*$ has order $n\,$ which means that $g^n\equiv 1(\mathrm{mod}\ m)\,$ and $n\,$ is the smallest positive integer with this property. Show that $g^m\,$ has order $\frac{n}{\mathrm{gcd}(n,m)}$ . Where $\mathrm{gcd}\,$ is the greatest common divisor function.

Extra Credit This is a more general version in a group $G\,$ .
Let $g\in G\,$ with $o(g)=n<\infty\,$ . Then for every integer $m,\$
$o(g^m)= \frac{n}{\mathrm{gcd}(n,m)}$

Friday, Nov. 4, 2005

Show that if $a\,$ and $b\,$ are integers then $\frac{a^2-2}{2b^2+3}$ is not.

landen used Legendre symbols to solve this. There may be a more elementary solution.

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