3
$\begingroup$

It is now proved that, for integer $n\geq 2$, the equation $x^n+y^n=z^n$ has integer solution only when $n=2$.

When $n=2$, this equation has an infinity of solutions.

My question is whether there is some established knowledge on the density of solutions and its variation in the space of integer pairs $(x,y)$ or $(x,z)$.

The following was added after reading the first answer by 01000100 (thanks).

The Lehmer asymptotic evaluation does give some indication of what I am after. But it is limited. If, in the $(x,y)$ plane, all pythagorean triples were concentrated around the $x=y$ line (the first diagonal), none being near the horizontal and vertical axes, it could follow the Lehmer density result, though be far from asymptotically uniform in relative density.

So I would rather consider something more or less like the following (which may not be quite correct a way to express what I have in mind):

Let $P(m,n)$ be the number of primitive Pythagorean triples $(x,y,z)$ with $m/2\leq x\leq m$ and $n/2\leq y\leq n$, with $gcd(x,y,z)=1$. What is the limit $$\lim_{p \rightarrow \infty} \frac{P(ap,bp)}{abp^2/4}$$ where $a,b,p\in \mathbb N$? Note that $abp^2/4$ is simply the surface of the rectangle where primitive Pythagorean triples are considered for $P(ap,bp)$.

This is intended just to control the plane direction in which the density is measured. But one could consider more complex relations between the two parameters of $P$.

Or another question (which I think is not equivalent, but I am not sure):

Is the set $\{x/y\mid x^2+y^2=z^2 \wedge (x,y,z)\in \mathbb N^3\}$ dense in the reals?

And I have similar questions regarding $(x,z)$ rather than $(x,y)$.

$\endgroup$
  • $\begingroup$ I will forever wonder what in my original question motivated downvoting. I am not a mathematician, I know too little of the topic for effective search on my own, and my own motivations for the question are not mathematical. $\endgroup$ – babou Sep 27 '14 at 22:02
2
$\begingroup$

First, it's a little silly to call this "Fermat's equation".

Now, points with both coordinates rational are dense on the unit circle; any line through $(1,0)$ with rational slope hits the circle in such a point. This should enable you to answer your questions about density.

$\endgroup$
  • $\begingroup$ Thanks. Sorry for the name used. My interest came from an actual equation that happens to have the same form. I was so convinced that I know noting of the topic that it blocks reasonning. And I even misread your answer. OK now, so simple. $\endgroup$ – babou Oct 12 '14 at 15:10
3
$\begingroup$

Let $P(n)$ be the number of primitive Pythagorean triples, that is $\gcd(x,y,z)=1$, with $z \leq n$.

Lehmer showed that

$$\lim_{n \rightarrow \infty} \frac{P(n)}{n} = \frac{1}{2\pi}$$

The reference is Lehmer, D.N. (1900). Asymptotic evaluation of certain totient sums. American Journal of Mathematics, 22(4), 293-335.

$\endgroup$
  • $\begingroup$ A heuristic derivation of this result is given by Fred Helenius here. mathforum.org/kb/message.jspa?messageID=1673162 $\endgroup$ – Daniel Pietrobon Sep 27 '14 at 10:21
  • $\begingroup$ Thanks. I used your answer to mak my question more precise, though I already mentioned that I was more interested in pairs of arguments. Or can it be deduced simply from Lehmer's result? $\endgroup$ – babou Sep 27 '14 at 22:06
  • $\begingroup$ This is beautiful... $\endgroup$ – recursive recursion Sep 27 '14 at 23:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.