Junior 2010/5
Find all pairs of positive integers $(m,n)$ which satisfy $m^{3}+1331=n^{3}$
"Questions we can't answer are far better than answers we can't question"
Re: Junior 2010/5
Edited. If you are using proper latex code why are you not putting \$ $ around it?
This problem is a special case for (very) well known Fermat's last theorem (see here http://en.wikipedia.org/wiki/Fermats_Last_Theorem ). As it was finally proved in 1995, you can use it in problem solving (though it takes the fun out of it).
This problem is a special case for (very) well known Fermat's last theorem (see here http://en.wikipedia.org/wiki/Fermats_Last_Theorem ). As it was finally proved in 1995, you can use it in problem solving (though it takes the fun out of it).
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: Junior 2010/5
Solving without using Fermat's last theorem.
$n^3-m^3 = 1331$
$=(n-m)(n^2+nm+m^2)$
And,$1331 = 11^3 $
Now,we have two cases,
$(1).(n-m)=11, (n^2+nm+ m^2)=11^2 $
But if, $(n-m)=11, then 11^2=(n^2-2mn+m^2)$ and thus, case $1$ is obviously not true.
$(2). (n-m)=1, (n^2+nm+m^2)= 11^3 $
Then,$(n-m)^2+3mn = 11^3$
or,$1+3mn=1331 $
or,$ 3mn=1330 $
or,$ mn=1330/3 $
But here, $mn$ has no positive integer solution.
So, we can say that $ m^3+1331=n^3 $ has no positive integer solution for $(m,n)$
$n^3-m^3 = 1331$
$=(n-m)(n^2+nm+m^2)$
And,$1331 = 11^3 $
Now,we have two cases,
$(1).(n-m)=11, (n^2+nm+ m^2)=11^2 $
But if, $(n-m)=11, then 11^2=(n^2-2mn+m^2)$ and thus, case $1$ is obviously not true.
$(2). (n-m)=1, (n^2+nm+m^2)= 11^3 $
Then,$(n-m)^2+3mn = 11^3$
or,$1+3mn=1331 $
or,$ 3mn=1330 $
or,$ mn=1330/3 $
But here, $mn$ has no positive integer solution.
So, we can say that $ m^3+1331=n^3 $ has no positive integer solution for $(m,n)$
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
- Charles Caleb Colton
- Charles Caleb Colton
-
- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: Junior 2010/5
According to the Farmat's last theorem no three positive integers $a, b,$ and $c$ satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than $2$.
Given equation,
\[m^3+1331=n^3\]
\[m^3+11^3=n^3\]
Where $m$ & $n$ both are positive integer.
So,according to the Farmat's last theorem ,
we can say that $m^3+1331=n^3$ has no positive integer solution for $(m,n)$