Question

not a question
Art of Problem Solving
Methods of Solving Diophantine Equations

1. Discriminant of quadratic equation

Answer 1

If the diophantine equation given is a quadratic equation with 2 variables, the discriminant can be very useful to determine the possible values of the variables.
If the variables are stated to be positive integer, it suggests that the discriminant must be a perfect square (or in rare cases, the square of a rational number).
----------------------------------------------------------------------------
Here’s a problem (\(\color\red{\text {by myself}}\)) to introducing this method :

Problem: Find all pairs of positive integers \(\color\red{\text {(m,n)}}\) such that : \(\color\red{\ { m+n=m^2-3mn+2n^2}}\)


We have to arrange the equation to appear as a quadratic equation in terms of \(\color\red{\text {m}}\) or \(\color\red{\text {n}}\) .i do it for \(\color\red{\text {m}}\) : Upon rearrangement we have\(\color\red{\ {m^2-(3n+1)m+2n^2-n=0 }}\). By taking discriminant, we obtain \( \)\(\color\red{\ { \Delta=n^2+10n+1}}\).
So the discriminant must be a perfect square but we have:

\( (n+1)^2< \Delta=n^2+10n+1<(n+5)^2\)
Hence:

\( n^2+10n+1=(n+2)^2 , (n+3)^2 \ and \ (n+4)^2\)

from this we get \(\color\red{\text {n=2}}\) and putting \(\color\red{\text {n=2}}\) in original equation gives \(\color\red{\text {m=1}}\) so the only solution is \(\color\red{\text {(m,n)=(1,2)}}\)
----------------------------------------------------------------------------

Answer 2

@mukushla

Its very nice method

Answer 3

Excellent...

Sorry to say but I am not getting how did you find n from the last equation..??
Can you explain it to me??
@mukushla

Answer 4

Yeah I got it from (n+3)^2..

Answer 5

(n+1)^2=n^2+2n+1<n^2+10n+1<n^2+10n+25=(n+5)^2

so we have 3 options for n^2+10n+1 to be a perfect square
n^2+10n+1=(n+2)^2 --------> n=1/2 not a integer
n^2+10n+1=(n+3)^2 --------> n=2
n^2+10n+1=(n+4)^2 --------> n=15/2 not a integer

Answer 6

must say that : great tutorial ..
\[\Huge{\text{Great}\mathbb{Job!}\textbf{Keep}\mathbb{It}\textbf{Up}}\]

Answer 7

@mathslover
tnx my friend...

Answer 8

No thanks , u were the desrver

Answer 9

I wish I was good at maths like you..I'm so dumb :(

Answer 10

\[\LARGE{Great \space job \space Dude \space ;)}\]

Answer 11

I wish there was a search function on this site :-D gj m8!

Answer 12

Superb!!

Answer 13

great job! i used such things before,but never found it anywhere so clear!

Answer 14

@quarkine
happy to hear that

Answer 15

i hope u give me permition to save this link... thanks

Answer 16

sure...

Answer 17

nice :D