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April 5th, 2012, 01:38 AM   #1
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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.
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April 5th, 2012, 01:17 PM   #2
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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 mn 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.
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April 6th, 2012, 03:35 AM   #3
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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.
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April 17th, 2012, 07:27 AM   #4
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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.
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November 21st, 2013, 10:12 PM   #5
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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.
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November 22nd, 2013, 06:44 PM   #6
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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.
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November 28th, 2013, 10:39 PM   #7
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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
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November 29th, 2013, 06:42 AM   #8
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Re: Formula for all and only primitive Pythagorean triples.

Sorry, now nothing for Re.
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December 12th, 2013, 11:56 AM   #9
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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/
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December 13th, 2013, 07:57 AM   #10
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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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