An OLD NT

For discussing Olympiad Level Number Theory problems
Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm
An OLD NT

Unread post by Asif Hossain » Thu May 06, 2021 10:42 pm

Find all positive integers $x,y,z$ such that $x^2 +y^2 =3z^2$.
Hmm..Hammer...Treat everything as nail

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: An OLD NT

Unread post by Asif Hossain » Fri May 07, 2021 2:50 pm

Asif Hossain wrote:
Thu May 06, 2021 10:42 pm
Find all positive integers $x,y,z$ such that $x^2 +y^2 =3z^2$.
Here is a elegant solution (saw in thanic bhaiya's book):

It is easy to check that $3|x,y$ write $x=3a$ and $y=3b$ now it follows that $3|z$ writing $z=3c$ implies
$3a^2 +3b^2= 9c^2 \Rightarrow a^2+b^2=3c^2$
so if $x,y,z$ is a solution then $x/3,y/3,z/3$ is also a solution. Iterating this we get an infinte descent in the set of positive integers.
CONTRADICTION.SO NO SOLUTIONS for $x,y,z$ $\square$
Hmm..Hammer...Treat everything as nail

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