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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 12,944

Euclid had proved that there are an infinite number of Pythagorean Triples.

The proof just has three or four statements, it is elegant, logical and flawless. Can any member/Mod post the proof here? If none does, I shall post the proof within a week.

Before coming across the proof, I used a different rationale.

The proof is something like this and begins with the simplest Pythagorean triple, viz.

(3,4,5).

Let n be any natural number from 1 to ∞.

Lets try if the set of three numbers, 3n, 4n, and 5n satisfies the condition of being a Pythagorean triple.

(3n)² +(4n)² = 9n² + 16n² = 25n² which is the square of 5n!

Hence, (3n, 4n, 5n) always form a Pythagorean triple for any natural number from 1 to ∞. Hence there are an infinite number of Pythagorean triples! eg (6,8,10), (9,12,15), (12, 16, 20) etc.

I concede the proof is neither elegant nor interesting enough. But then, the most import point is *q.e.d*.

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 12,944

**Euclid's Proof**

*In my own words....*

1. There are an infinite number of natural numbers 1,2,3,4,5,..............................n.

2. There are an infinite number of squares of these natural numbers, 1², 2², 3², 4², 5².......n²

3. The successive squares of these natural numbers differ by consecutive odd numbers, viz. 1, 3, 5, 7, 9.....etc. and the number of such consecutive odd numbers is infinite, from (1) and (2) above.

4. Some of these differences are themselves squares, like 9, 25, 49, 81, 121 etc. which form a fraction of the number of elements in (3) above. But, fraction of an infinity is also infinity.

*Therefore, there exist an infinite number of Pythagorean triples.*

Character is who you are when no one is looking.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Great observations, but I would hardly call that a proof. More a sketch or an outline. Of course, a much more sophisticated proof will show that

[align=center]

[/align]Is not only always a Pythagorean triple for any m and n (hence, there are an infinite number of them), but all primitive Pythagorean triples may be described in this way.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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