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Pythagorean Soliloquy
by Total Gadha - Wednesday, 31 December 2008, 02:07 PM

cat 2009 cat 2010 pythagorean triplets mba 2009Years ago, my Italian language teacher and my classmates learning Italian used to question me how I could pick up the language so fast. “Perche sono un insegnante di matematica,” was my standard answer. That would leave them baffled. What was the relation between language and mathematics? Little do my classmates or many other students, especially our CAT aspirants, understand that language and mathematics are the two sides of the same coin; both consist of symbols connected through logic. And more often than not, even the logic they follow is the same. The symbols for both were gradually developed. The mathematician Euler was responsible for our common, modern-day use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of –1, Π for pi, Σ for summation. Shakespeare is credited with coining nearly 1700 words in the English language. These are our symbols. The logic was also developed gradually; the grammar, mathematical proofs, critical reasoning, logical reasoning, etc. form the parts of logic that bind those symbols. Therefore, if you are a logical creature, chances are that you would not face difficulty understanding either of these subjects, provided you are comfortable with the symbols- English vocabulary or mathematical notations, whatever the case maybe.

The progression of thoughts of a writer is not markedly different from that of a mathematician. While reading a good passage one can automatically guess the next paragraph. In the same manner, a mathematician proceeds from one thought to another in a logical manner, removing obstacles in his path to discovery and finding answers to the problems one after another.

cat 2009 pythagorean triplets..

cat 2009 pythagoras triplets

Re: Pythagorean Soliloquy
by Gowri Nandana - Wednesday, 31 December 2008, 02:22 PM
  Awesome article........
Re: Pythagorean Soliloquy
by Gowri Nandana - Wednesday, 31 December 2008, 02:35 PM

Happy New year to the super TG team.

May God gift u with 48 hours a day to manage ur wonderful site


Re: Pythagorean Soliloquy
by Dagny Taggart - Thursday, 1 January 2009, 05:28 PM
  A very Hppy New Year to you too, Gowri. smile
Re: Pythagorean Soliloquy
by Gul Gul - Thursday, 1 January 2009, 07:19 PM
  One of the best ....!
Re: Pythagorean Soliloquy
by sandeep somavarapu - Thursday, 1 January 2009, 08:55 PM
  Sir ji...What an explantion sir ji.....
------------one concept can change your thought process
I wish you and TG family a very happy New year.
Re: Pythagorean Soliloquy
by rajat shukla - Friday, 2 January 2009, 08:28 PM
  happy new all of u..
Re: Pythagorean Soliloquy
by rashi agarwal - Friday, 2 January 2009, 11:59 PM
  hello TG sir,
 A very happy New Year to you...Great article......

 sir, i hav a little confusion.To calculate the no. of right triangles with a given length N as one of its leg,
N= 2a x pb  x qc x .................

The formula is [(2a -1)(2b+1)(2c+1)....-1 ]/ 2

But sir if we want to calculate with 25 as the length of the leg. then a =0. then whole term will be -ve. this method will not be used for the odd numbers..please sir correct me if I am wrong.

Re: Pythagorean Soliloquy
by Total Gadha - Saturday, 3 January 2009, 02:18 AM
  Hi Rashi,

Don't consider the term 2a - 1 if N is an odd number, i.e. no power of 2 is present.

Total Gadha
Re: Pythagorean Soliloquy
by Dipanjan Biswas - Sunday, 4 January 2009, 11:48 PM
  Thank you sir a lot for such a nice article..It really fosters neural network of thoughts..Simultaneously it covers up number system...Thanks once again..
Happy New Year to all of TG family
TG rockzzz

Re: Pythagorean Soliloquy
by Tuhin Banerjee - Monday, 5 January 2009, 06:24 PM

Hi TG,

Really a awesome article.

Thanks a zillion mam.

Re: Pythagorean Soliloquy
by gadha abc - Monday, 5 January 2009, 06:49 PM
  Each sentence speaks the intelligence..
Happy new TG sir and Mam.

Sir, can u please little elaborate how that formula u derive N+5C5 for the dice problem.

Re: Pythagorean Soliloquy
by himanshu mishra - Sunday, 11 January 2009, 10:49 PM


          This article is superb...........all the comments are useless for this nugget of wisdom.  

Re: Pythagorean Soliloquy
by Sharadha Kuntumalla - Monday, 26 January 2009, 12:57 AM

Hello TG Sir,

I want to copy the article in a word document so that I can print it, highlight important points and keep revising it. But the content is not getting aligned properly on the word document. I won't find this difficulty when I am copying verbal articles/exercises. Can you please help me with a printable version of these wonderful articles.


Re: Pythagorean Soliloquy
by Total Gadha - Monday, 26 January 2009, 11:09 AM

Hi Sharadha,

Many of these articles are images. Right click on them, 'copy' them and paste them in paint and then save them as images. Then you can print them out.

Total Gadha

Re: Pythagorean Soliloquy
by sumit jamwal - Thursday, 29 January 2009, 12:50 AM
  hi sir,
as you stated in the article
if i take 25 As hypotenuse

then there are


so  ((2*2+1)-1)/2 = 2
bt i cud find only

5^2=3^2  +4^2

confused  sad
Re: Pythagorean Soliloquy
by Total Gadha - Thursday, 29 January 2009, 10:25 AM

Hi Sumit,

You are taking 5 or 25 as hypotenuse? If you are taking 25, then you need to take 625 = 252.

Total Gadha

Re: Pythagorean Soliloquy
by Anurag Rai - Sunday, 1 February 2009, 02:06 PM
  Nice and Brilliant smile
Pythagorean Soliloquy
by Abhishek Bansal - Monday, 2 February 2009, 02:17 AM
  As i was attending a class on Geometry. I was told that there there is one triplet such as (20,21,29). I started searching and wanted to find a way to get to the number of triplets if i am given one of the sides.

A side can be the longest side (i.e. hypotenuse) or base/perpendicular.

i will say its by sheer coincidence, that someone told me about this site today when i was taking the class - and in that class i took the case of one of the sides being 5^11.

i knew a way to find when 5^11 is not the longest side or any number for that matter when it is not the longest side..

  if we are looking for integral solutions then (a-b)*(a+b) should be even or odd...

so 11 triangles are possible.

I wrote a code in Matlab (just to ensure that i am not the one doing the calculation) - i took 5^(n) ... n starting from 1,2,3,..... 12,13...

at n = 15 it started showing some errors related to memory.

I really need to check the code, and the results.. because at n  =12 it gives me 16 triangles which are possible when 5^12 is the hypotenuse and 5^13 gives me 60 triangles.

what i saw till 5^11.. the answer will be the power itself.

and 3^n and 7^n won't give me any triangles.

and 31^n and 37^n.. will start giving me triangles for some value of n > 2,3

so can u check the validity.

I can be wrong. i just have my code and the results and those i will post tomorrow.

Re: Pythagorean Soliloquy
by Abhishek Bansal - Monday, 2 February 2009, 02:42 AM
  It is quite possible that the number suddenly increases because of the precision errors.
Re: Pythagorean Soliloquy
by Harish Bansal - Wednesday, 4 February 2009, 12:22 AM

Hi TG,

I tried to derive the same formula that u have written at the top by using the same example of 60^2. I got a different result.

My result is: {(a-1)(b+1)(c+1)(d+1)....-1}/2

Please clarify

Re: Pythagorean Soliloquy
by sumit jamwal - Wednesday, 4 February 2009, 08:44 AM
  Thanks TG mistake
Re: Pythagorean Soliloquy
by Total Gadha - Wednesday, 4 February 2009, 09:25 AM

Hi Harish,

Maybe if you can explain how you derived the formula, I can point out the mistake.

Total Gadha

Re: Pythagorean Soliloquy
by alsadra @TG - Monday, 16 February 2009, 04:01 PM
  sir i 've been reading this article for more than 90 minutes. had to imagine a lot. can u please explain me how u calculated the no of ways of expressing 65 as a sum of two squares towards the end of the article.

the article is very informative and very dense in information. thanks sir...
Re: Pythagorean Soliloquy
by srihari sankararaman - Thursday, 19 February 2009, 05:49 AM

Hello Sir,

              Sheer brilliance... I was not able to prove the last theorem. Could you tell me how you came to this conclusion.(n+5)c5?

Once again,Kudos to TG.

Re: Pythagorean Soliloquy
by syed haque - Wednesday, 25 February 2009, 05:53 PM

Hi TG,

Can you please explain why you have divided the number of factors by 2?

As two different pairs can give different values of a and b and will result in two right triangles.Sir please explain me this.


Re: Pythagorean Soliloquy
by amrit jajodia - Sunday, 1 March 2009, 12:28 AM
Just one word for the article ..... Awesome!!!!!

I am not clear about the no of distinct outcome formula.
Just for the case if N = 2 then no of instinct outcome possible is nos from 2 to 12. that is 11 outcomes but with the formula i get 21. Can you please clarify on this.

And moreover can you please put up the answer for those three questions also.
Re: Pythagorean Soliloquy
by Abhishek Bansal - Sunday, 1 March 2009, 10:46 AM
  you are talking about the sum... when you see 2 to 12....

see the total number of outcomes when you throw two dice will be 36 if the dice are not identical to each other....

1,2 is different from 2,1

but when you throw two identical dice together....

1,2 is same as 2,1

there is no difference as you won't be able to distinguish...

so those kind of cases will be 21...

1,1 2,2 3,3 4,4 5,5 6,6 if you take this cases out of the sample space.. there are 30 more cases left... now these 30 cases have (p,q) and (q,p) where p is not equal to q.

so these cases are counted twice....

30/2 + 6 = 15 +6 = 21cases
Re: Pythagorean Soliloquy
by yogesh bansal - Tuesday, 23 June 2009, 02:47 PM

hi...answer of first 2 questions out of last 3 questions...

1.) 48

2.) 27

Am i correct???

Tg plz let me knw..

Re: Pythagorean Soliloquy
by Mohit Bhambri - Friday, 4 September 2009, 10:06 PM
1) 48
since we have to get 10 distinct
[(2a-1)(2b+1)...-1]/2 =10 which gives (2a-1)(2b+1)...= 21
let it be 3 and 7 taking 7 for powers of 2 and 3 for powers of 3
we get 2^4*3 =48 ..pls confirm

Re: Pythagorean Soliloquy
by Mohit Bhambri - Friday, 4 September 2009, 10:52 PM
2) I too think it is 2^7. For a number to be hypotenuse it must have a prime no of form 4n+1 in its factors. So if we start with a number of form 5x(..)
so we have to get  remaining 5 triangles from side as 5x(..) which means
[(2a-1)(2b+1)(2(1)+1)-1 ]/2 =5  cause we already let 5 be a factor thats y the 2(1)+1
*if there is a power of2
which gives 3x something =11 not possible hence having a single 4n+1 factor is ruled out
similarly having 2 4n+1 factors is ruled out i.e 13, 5 at the same time therefore only possibility to have 6 triangles is with it as SIDE ONLY not hypotenuse.

by  logic of 1) we get N= 2^7
Correct me if im wrong
Re: Pythagorean Soliloquy
by Mohit Bhambri - Friday, 4 September 2009, 10:54 PM
There is some problem in framing question 3 as it doesnt specify 3 traingles having integral sides of equal areas.
Re: Pythagorean Soliloquy
by srinivasan ravi - Friday, 11 September 2009, 12:20 AM
  hi mohit,
can u explain the 2nd question more clearly..thanks..
Re: Pythagorean Soliloquy
by Raju Singh - Monday, 28 September 2009, 02:34 AM
  @ Mohit

Could we hav some simple approach for sol of 2 Question asked:
Smallest number for 6 distinct right triangle for integer sides:


As RHS is 13 which is prime number so any of LHS term must have 13 as one of the number.

For number to be minimum only (2a-1) must be equal to 13 and rest all terms as 1
(2a-1)=13, (2b+1) =1 and (2c+1)=1 etc..
hence a=7 and b=c=...=0
And thereby 2^7 as the smallest number for rt triangles with integer side lengths.
Mohit what u hv elaborated I'm not getting plz clarify it.
Re: Pythagorean Soliloquy
by Deepak Kumar - Saturday, 9 October 2010, 12:31 PM

Hi TG Sir,

Really a great artical. I liked the way you have written the artical, logically coming to the next step, the way you told in the preface of the artical.


Re: Pythagorean Soliloquy
by abhishek abhi - Saturday, 1 October 2011, 08:46 PM
  sir in the first problem ,,to find out odd pairs 3*3 is done ,,sir could you please explain me how to find odd pairs in those factor pairs
Re: Pythagorean Soliloquy
by Just Gadha - Wednesday, 5 October 2011, 10:58 AM
  How do u get the total no. of solutions of the equation
as N+5
Re: Pythagorean Soliloquy
by Kamal Joshi - Friday, 7 October 2011, 05:40 AM

Please explain, how did you derive (2^2+1^2)(2^2+3^2)(2^2+1^2)(2^2+3^2) = 63^2+16^2 = 33^2+56^2???
Re: Pythagorean Soliloquy
by Kamal Joshi - Friday, 7 October 2011, 04:22 PM
  Ok got it, sir!!! You derived different combinations of 65^2 by (a^2+b^2)(c^2+d^2)= (ac+bd)^2+(ad-bc)^2 = (ac-bd)^2+(ad+bc)^2. Correct?
Re: Pythagorean Soliloquy
by prabhat taneja - Wednesday, 17 October 2012, 05:38 PM
  one word.. SUPERB.. I liked the article overall, but the footnote made my day.. Thanks to Mr Mahajan for the unique solution and TG for such fabulous a post. smile