316 reputation
18
bio website leonettipaolo.wordpress.com
location Milan, Italy
age 26
visits member for 1 year, 9 months
seen Apr 15 at 17:55

May
9
awarded  Yearling
Dec
30
suggested suggested edit on Diophantine equation: $x^2+y^2+z^2=n(xy+yz+zx)$
Dec
28
comment Diophantine equation: $x^2+y^2+z^2=n(xy+yz+zx)$
Oh, wow. I cannot add more than a vote for your time and patience, thank you!
Dec
28
revised Diophantine equation: $x^2+y^2+z^2=n(xy+yz+zx)$
Two typos, $X^2 \equiv 2-3$ to $X^2 \equiv -3$ and the equation of $C$..
Dec
28
suggested suggested edit on Diophantine equation: $x^2+y^2+z^2=n(xy+yz+zx)$
Dec
28
comment Diophantine equation: $x^2+y^2+z^2=n(xy+yz+zx)$
@YongHaoNg Your completion of squares is a known technique, but here it looks magic. I checked term by term, and it is correct. Can you show us in detail how you obtained it?
Nov
5
answered Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?
May
21
awarded  Critic
Apr
3
comment Finding all integer solutions of $5^x+7^y=2^z$
That is almost the same of mine, well done: +1.
Apr
3
comment Finding all integer solutions of $5^x+7^y=2^z$
@Ivan Loh, I am going to check your one in a few :)
Apr
3
comment Finding all integer solutions of $5^x+7^y=2^z$
@Potato: I already posted my solution on my blog..
Apr
2
comment Finding all integer solutions of $5^x+7^y=2^z$
Indeed it was assumed implicitely that $x\ge 1$; in rough terms, half problem is missed, i.e. all cases $x\le 0$; except that, we are talking about trivial cases.. I am still waiting if someone finds a elementary solution different from mine
Apr
2
comment Diophantine equation $x^2 + 32x = y^3$
That's ok :) Cheers Macavity!
Apr
2
answered If $a$ and $b$ are integers such that $9$ divides $a^2 + ab + b^2$ then $3$ divides both $a$ and $b$.
Apr
2
comment Diophantine equation $x^2 + 32x = y^3$
Sadly, there is still a mistake :/ the equation $$2v(v+1)=w^3\text{ with }2\mid v+1$$ implies only that $v$ is a cube such that $v\mid w^3$, i.e. $$2(v+1)=\left(\frac{w}{\sqrt[3]{v}}\right)^3.$$ In practice we know that $\text{rad}(v) \mid \text{rad}(w)$ but we cannot conclude that $v\mid w$.. Anyway, I like that you listen all comments, you have my thumb up too, although I still didnt realize why it should be useful :)
Apr
2
comment Finding all integer solutions of $5^x+7^y=2^z$
Thumb up. You got the non-elementary solution that I did know, but really in a few. Just a thing to avoid a reference: you can skip the Ramanujan equation, just looking in sequence at modules $25$, $32$ and $17$. Just to complete, at this point if $\min\{x,y,z\}<0$ then also $\max\{x,y,z\}<0$, and getting immediately a contradiction.
Apr
2
comment Finding all integer solutions of $5^x+7^y=2^z$
As far as I know, the ABC conjecture is still not even officially accepted, so.. Being more explicitely, with "elementary" I mean that a high school student can completely solve it. Although that, "elementary" does not mean that the question is "simple": noone from the online contest managed to solve it, and I took a whole week to do it
Apr
2
revised Finding all integer solutions of $5^x+7^y=2^z$
deleted 13 characters in body; edited title
Apr
2
awarded  Commentator
Apr
2
comment Diophantine equation $x^2 + 32x = y^3$
I am glad, you realized the problem now ;)