Q) Let \(a,b,c\) be three natural numbers such that \(a<b<c\) and \(\text{gcd}(c-a,c-b)=1\). Suppose there exists an integer \(d\) such that \(a+d,b+d,c+d\) form the sides of a right-angled triangle. Prove that there exist integers \(l,m\) such that \(c+d=l^2+m^2\)
Before we go to the solution of this problem, let us get into some theory of Pythagorean Triplets. We define a triplet of integers \((a,b,c)\) as a Pythagorean triplet, if \(a,b,c\) form the sides of a right angled triangle. So if \(a<b<c\), we must have \(a^2+b^2=c^2\). Without loss of any generality we can assume that \(\text{gcd}(a,b,c)=1\).
We can basically generate pythagorean triplets. Set \(a=s^2-t^2\), \(b=2st\) and \(c=s^2+t^2\) for positive integers \(s,t\) with \(s>t\). From this substitution we see that a pythagorian triplet can be generated for any \(s,t\). Needless to mention that \(\text{gcd}(s,t)=1\). And hence, there can be infinitely many pythagorean triplets. Also all pythagorean triplets are of the said form. Thus, for example, if we consider the pythagorean triplet \((3,4,5)\), we have \(s=2,t=1\).
For the problem at hand, given that \(\text{gcd}(c-a,c-b)=1\). We claim that \(\text{gcd}(a,b,c)=1\), otherwise, if \(\text{gcd}(a,b,c)=k\), where \(k \ne 1\), then we have \(\text{gcd}(c-a,c-b)=k \ne 1\). This also shows that \(\text{gcd}(a+d,b+d,c+d)=1\).
Given that \(a+d,b+d,c+d\) form the sides of a right angled triangle, which means that they form a Pythagorean triplet. Thus, we have \(c+d=s^2+t^2\) for some integers \(s,t\).
We call a triplet \((a,b,c)\) as a primitive Pythagorean triplet if they are coprime. Can you show that they are also relatively coprime, i.e \(\text{gcd}(a,b)=\text{gcd}(b,c)=\text{gcd}(c,a)=1\)?
Let’s look at another problem that uses Pythagorean triplets. Consider the Arithmetic progressions of 3 numbers – all of whom are perfect squares. Can we find 1 such AP?
It turns out that there are infinitely many such triplets \((a,b,c)\) such that \(a^2,b^2,c^2\) are in AP. To prove this, we need to show that the equation \(a^2+c^2=2b^2\) has infinitely many solutions in integers.
Let’s consider a Pythagorean triplet \((u,v,w)\). We thus know that \(u^2+v^2=w^2\).
Set \(a=u+v\) and \(c=|u-v|\).
Thus \(a^2+c^2=(u+v)^2+(u-v)^2=2(u^2+v^2)=2w^2\). Since we know that there are infinitely many Pythagorean triplets, we can say that the equation \(a^2+c^2=2b^2\) has infinitely many integer solutions.
Test Your Concepts:
Show that there are infinitely many rational points that lie on the circle \(x^2+y^2=2\).
Ι love reading an article that cann make people think.
Nice way to articulate the concepts, but looking forward to more such advanced mathematics stuff from the daily tutors.
I think a little bit of diophantine equations’ theory would have helped in this regard.
I want soln of test ur concpts.. where see them???
where can see the solution of the ‘test ur concpts’??
Hey there, I am into Mathematics tutoring as a hobby for high school pupils. Looks good to see people from India taking up Mathematics as an interestiing course to pursue. Would be great if you can cover College Mathematics on top of these.
But anyway Thanks for your personal marvelous posting! I seriously enjoyed reading it,
you’re a great author.I will be sure to bookmark your blog and will come back at some point.
I want to encourage you continue your great posts, have a nice holiday weekend!
Sorry, but don’t you think you can start with some elementary concepts? I’m just starting Olympiad prep and this sounds too technical for me 🙁
How to prepare for rmo?
Do you provide online tutoring of any kind? Even over video calls if possible? I am from Calcutta and would like to appear for RMO this year. Also if you tutor, what’re your fees?
Can you solve rmo 2007 paper for me?
Can you give the solution of the full paper?
Can you upload the last qustion’s solution?
Solution for the ratinal points questions?
Could you post some good ebooks to learn such and similar concepts for the IMO?
Just to say a ‘hi’ from Calcutta – nice job covering the concepts of the Olympiads!
I could not find any more such concepts core to the RMO being covered over here. Am I missing something or is there more coming like this?
Not sure if you read the comments, but a lot of us would appreciate if you can post the solutions to the questions that you post for practice as well.
Impressive, though would like to have the solution for the question of rational points if possible
I know that this is quite handy and all, but haven’t seen much of IMO’s questions to be based on Pythagorean triplets.
Wonder how frequently this topic is visited in the RMO/IMO Exams.
Hi, Do you mind posting a few more questions for CMI Entrance?
Don’t know how difficult it’dbe, but do you mind starting some videos on this? I can help if need be, but a Youtube channel would be great to have.
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up for the great info you have got right here on this post.
I will be coming back to your site for more soon.
Solutions for the exercises would be highly appreciated!
Another beautiful piece from the Daily Tutor. Subscribed!
Appreciate the effort behind to put up such a comprehensive tutorial, and so nicely explained.
Rarely seen anything more comprehensive on Pythagorian triplets so fsr on the web.
This RMO question actually looked a bit to trivial to me, if you compare this to the other questions of the exam.
This tutorial on Pythagorian triplets is easily one of the best that I could find in recent times!
Nice materials for Mathematics.
Long time supporter, and thought I’d drop a
comment.
You had earlier said about opening a forum for maths, where’s that Sir?
I was trying to proe both c and d are perfect squares, any idea if that’d work?
Bhai itna toh na ho payga hall mein, kuch bhi karlo
Hate these problems, unless you know it you cannot solve it 😐
Got this problem in the rmo 2007 exam, couldn’t solve 🙁
Pythagoras and his triplets – messing with our brains since eternity lol 😀