are there any positive integers k,a, b such that this equation is satisfied:
Yes, for example k = 9, a = 4, b = 8.
thanks, how do i go about proving that infinite solutions exist?
Here's the solution.
Then choose b such that b is even.
This implies b^2+1 is odd. If b^2+1 is a prime, then put k+a=b^2+1 and k-a=1. You will get k&a. If b^2+1 is not a prime, then choose k+a and k-a as its two odd factors. Solving, you get k & a.
As there is solution for all b>0 and b even, there are infinitely many solutions.
You can see that Orthodontist's solution is also one of these.
the case with 1 is a special case of
a good question that might arose from this is what number is greater: the number of pythogrean triplets or the above qudroplets?
neither, they are both the same (countably infinite); it is not a difficulct question, and has indeed already been answered in this thread where it is asserted that there are infintely many solutions to
why should [itex]b[/itex] be even?
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