My Math Forum Formula for all and only primitive Pythagorean triples.

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 April 5th, 2012, 12:38 AM #1 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Formula for all and only primitive Pythagorean triples. Formula for all and only primitive Pythagorean triples. 1. From only several Pythagorean triples: 4, 3, 5; 8, 15, 17; 12, 5, 13; 12, 35, 37; 20, 21, 29; 24, 7, 25; 28, 45, 53 and 36, 27, 45; I synthesize a formula, that may be you have never seen: $[2p(p+2q-1)]^{2}+[4pq+4q^{2}-2p-4q+1]^{2}=[4pq+4q^{2}-2p-4q+1+2p^{2}]^{2}$ (P) (p, q are intergers) 2. By means of re-marking the first time, with m = p, n = p + 2q -1 and from here, we find n > m and n is an even number when m is odd, and on the contrary, n is an odd number when m is an even. Therefore (P) is written: $(2mn)^{2}+(n^{2} - m^{2})^{2}= (n^{2} + m^{2})^{2}$ (P1) (n > m, n and m are natural numbers: 1, 2, 3 ... but when n is an even, m is an odd or when n is an odd, m is an even) 3. By means of re-marking the second time, with i = 2q -1, j = 2q -1 +2p and from here, we find j > i and i, j are odd numbers.Therefore (P) is written: $(\frac{j^{2}-i^{2}}{2})^{2}+(ij)^{2}=(\frac{j^{2}+i^{2}}{2})^{2 }$ (P2) (j > i and i, j are odd numbers) 4. Formula for all and only primitive Pythagorean triples: $(\frac{j^{2}-i^{2}}{2})^{2}+(ij)^{2}=(\frac{j^{2}+i^{2}}{2})^{2 }$ i, j are odd co-prime numbers and j > i. Particularly when i, j are co-prime we obtain all primitive Pythagorean triples.
 April 5th, 2012, 12:17 PM #2 Global Moderator   Joined: May 2007 Posts: 6,787 Thanks: 708 Re: Formula for all and only primitive Pythagorean triples. Euclid's formula[1] is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with m > n. The formula states that the integers a = m^2 - n^2 , b = 2mn , c = m^2 + n^2 form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and m–n is odd. If both m and n are odd, then a, b, and c will be even, and so the triple will not be primitive; however, dividing a, b, and c by 2 will yield a primitive triple if m and n are coprime.[2] Above from Wikipedia. You left out the case where (in formula 4) where i or j is even.
 April 6th, 2012, 02:35 AM #3 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Formula for all and only primitive Pythagorean triples. Euclid's formula generates all primitive triples and some non-primitive, but there are non-primitive triples that his method won't generate. I find this interesting so i share it here.
 April 17th, 2012, 06:27 AM #4 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Re: Formula for all and only primitive Pythagorean triples. I want to aim that the four items paper is very general and the forth formula (with i and j are odd co-prime intergers) is a Formula for all and only primitive Pythagorean triples.
 November 21st, 2013, 09:12 PM #5 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Re: Formula for all and only primitive Pythagorean triples. mathman wrote: "You left out the case where (in formula 4) where i or j is even." You can see the 3rd item of my piece, i and j are not even.
 November 22nd, 2013, 05:44 PM #6 Senior Member     Joined: May 2013 From: EspaÃ±a Posts: 151 Thanks: 4 Re: Formula for all and only primitive Pythagorean triples. Hello. You look at this, for if it him can be useful: http://www.mymathforum.com/viewtopic.php?f=40&t=40853 Regards.
 November 28th, 2013, 09:39 PM #7 Banned Camp   Joined: Dec 2012 Posts: 1,028 Thanks: 24 Re: Formula for all and only primitive Pythagorean triples. As probably already written here I found more interesting that: Given an odd integer A there are ALWAYS two integer B and C that satisfy: A^2 = C^2-B^2 If you keep: A, B= (A^2-1)/2, C= (A^2+1)/2 so you can see all triplets sorted by A odds. To obtain triple for ALL integers you've just to ADD the even 2*A values where A is what above. Thanks Ciao S.
 November 29th, 2013, 05:42 AM #8 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Re: Formula for all and only primitive Pythagorean triples. Sorry, now nothing for Re.
 December 12th, 2013, 10:56 AM #9 Newbie   Joined: Dec 2013 Posts: 3 Thanks: 0 Re: Formula for all and only primitive Pythagorean triples. Hi If you would like to read something more interesting and up to date on PPT's try these ... http://curiousmathsknols.com/article/fr ... to-fermat/ and http://curiousmathsknols.com/article/py ... -equation/
 December 13th, 2013, 06:57 AM #10 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Formula for all and only primitive Pythagorean triples. Okay, let me guess, can you be John Morrison himself or just a reader of math knols?

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