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        riddles >> medium >> Integer triangle with 120 degrees
        
(Message started by: Christine on Apr 23^rd, 2013, 10:35am)

Title: Integer triangle with 120 degrees
Post by Christine on Apr 23^rd, 2013, 10:35am

This type of triangle can be generated by

a = m² - n²
b = 2mn + n²
c = m² + mn + n²

we note
a² + ab + b² = c²

If m = 2, n = 1 ---> a = 3, b = 5, c = 7

the smallest triangle with an angle of 120 degrees having all 3 sides with prime numbers.

Is it the only triangle with the three sides being prime numbers? If yes, how to prove it?

Title: Re: Integer triangle with 120 degrees
Post by towr on Apr 23^rd, 2013, 10:41am

[hide]Well, n has to be 1, otherwise b can't be prime (n divides 2mn + n^2). And m-n has to be 1, because otherwise a isn't prime (m-n divides m^2-n^2). So therefore m=2, n=1 is the only solution with prime numbers.

Provided all such triangles can be generated in this way.[/hide]

Title: Re: Integer triangle with 120 degrees
Post by Christine on Apr 23^rd, 2013, 11:13am

on 04/23/13 at 10:41:10, towr wrote:

~~Can n be a prime number > 2 ?~~