Re: Pythagorean Soliloquy  
Awesome article........ 
Re: Pythagorean Soliloquy  
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  
A very Hppy New Year to you too, Gowri. 
Re: Pythagorean Soliloquy  
One of the best ....! 
Re: Pythagorean Soliloquy  
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  
happy new year...to all of u.. 
Re: Pythagorean Soliloquy  
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  
Hi TG, Really a awesome article. Thanks a zillion mam. 
Re: Pythagorean Soliloquy  
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. Regards 
Re: Pythagorean Soliloquy  
hi,tg This article is superb...........all the comments are useless for this nugget of wisdom. 
Re: Pythagorean Soliloquy  
hi sir, as you stated in the article if i take 25 As hypotenuse then there are 25=5^2 so ((2*2+1)1)/2 = 2 bt i cud find only 5^2=3^2 +4^2 confused 
Re: Pythagorean Soliloquy  
Hi Sumit, You are taking 5 or 25 as hypotenuse? If you are taking 25, then you need to take 625 = 25^{2}. Total Gadha 
Re: Pythagorean Soliloquy  
Nice and Brilliant 
Re: Pythagorean Soliloquy  
It is quite possible that the number suddenly increases because of the precision errors. 
Re: Pythagorean Soliloquy  
Thanks TG ..my mistake 
Re: Pythagorean Soliloquy  
Hi Harish, Maybe if you can explain how you derived the formula, I can point out the mistake. Total Gadha 
Re: Pythagorean Soliloquy  
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  
hi...answer of first 2 questions out of last 3 questions... 1.) 48 2.) 2^{7} ^{Am i correct???} ^{Tg plz let me knw..} 
Re: Pythagorean Soliloquy  
There is some problem in framing question 3 as it doesnt specify 3 traingles having integral sides of equal areas. 
Re: Pythagorean Soliloquy  
hi mohit, can u explain the 2nd question more clearly..thanks.. 
Re: Pythagorean Soliloquy  
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  
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  
How do u get the total no. of solutions of the equation a1+a2+a3+a4+a5=N as N+5 C 5 
Re: Pythagorean Soliloquy  
Sir, 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  
Ok got it, sir!!! You derived different combinations of 65^2 by (a^2+b^2)(c^2+d^2)= (ac+bd)^2+(adbc)^2 = (acbd)^2+(ad+bc)^2. Correct? 
Re: Pythagorean Soliloquy  
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. 