How many Pythagorean triangles are there whose area is expressed by 9 different digits?
Provide the values for the smallest and the biggest of them.
The legs (x,y) of a primitive pythagorean triangle are x=2ab and y=a^2-b^2 where (a,b) are relatively prime and of opposite parity. All other PTs can be obtained from primitive values by multiplying x and y by any integer k.
The area is one-half the product of the legs or Area = k^2 * ab * (a^2 - b^2). This is always even since one of (a,b) is even. Also, looking at values of a and b mod 3 shows area is divisible by 3 as well.
The sum of 0,1, ... 9 is 45 which is divisible by 3 so the excluded digit in the sought for area can only be 0,3,6,9.
Posted by xdog
on 2013-09-17 17:56:04