Prime Numbers Proof

Date: 05/23/2005 at 09:13:32
From: Karl
Subject: Prime numbers

Prove that every prime number is the leg of exactly one right triangle 
with integer sides.

Date: 05/23/2005 at 16:17:42
From: Doctor Floor
Subject: Re: Prime numbers

Hi Karl,

Thanks for your question.

I suppose that you mean that the leg is not the hypothenuse of the 
right triangle, and that the prime is to be odd (greater than 2, that 
is).  I will call right triangles with integer sides Pythagorean in 
this answer.

Let us consider a prime p.

Of course there is at least one Pythagorean triangle with a leg of 
length p.  That follows from the sequence of differences of the 
sequence of squares being the odd numbers:

  squares     1 4 9 16 25  36  49  ....
  differences  3 5 7  9  11  13

So p^2, being odd, is the differences of say n^2 and (n+1)^2, and 
n^2 + p^2 = (n+1)^2.  Of course this way we find only one Pythagorean 
triangle with a leg of length p.

Now suppose that p is the leg of a second Pythagorean triangle, 
distinct from the one found before.  Then we may write:

  f^2 + p^2 = g^2

for some f and g, which are not consecutive.  Hence

  p^2 = g^2 - f^2 = (g-f)(g+f)

but as g-f and g+f are unequal and both greater than one, this 
contradicts with p being square.  And we have that there is no such 
second triangle.

If you have more questions, just write back.

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/