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

 Number Theory Number Theory Math Forum

 April 5th, 2012, 01:38 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: (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: (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: (P2) (j > i and i, j are odd numbers) 4. Formula for all and only primitive Pythagorean triples: 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, 01:17 PM #2 Global Moderator   Joined: May 2007 Posts: 6,856 Thanks: 745 Re: Formula for all and only primitive Pythagorean triples. Euclid's formula 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. Above from Wikipedia. You left out the case where (in formula 4) where i or j is even. April 6th, 2012, 03:35 AM #3 Math Team   Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 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, 07:27 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, 10:12 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, 06:44 PM #6 Senior Member   Joined: May 2013 From: España 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, 10:39 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, 06:42 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, 11:56 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, 07:57 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? Tags formula, primitive, pythagorean, triples ,

,

,

,

,

,

,

,

,

,

,

# triples of primes general formula

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post billymac00 Number Theory 19 March 19th, 2013 08:38 PM nonlinear Number Theory 10 October 3rd, 2011 11:03 PM Pell's fish Number Theory 3 August 5th, 2011 11:38 AM julian21 Number Theory 2 November 13th, 2010 01:01 PM julian21 Number Theory 3 November 12th, 2010 12:58 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      