Junior 2010/5
Find all pairs of positive integers $(m,n)$ which satisfy $m^{3}+1331=n^{3}$
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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).
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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.
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samiul_samin
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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 $(m,n)$