Prime Numbers ProofDate: 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/ |
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