This is a really nice problem. If the solution just crosses your mind, great! If not, keep thinking. I can assure you, when you find the solution you will feel real good.
Prove that there are an infinite integers that can't be written as $x^{\tau(x)}+y^{\tau(y)}$.
Here, $\tau(n)$ is the number of positive divisors of $n$, as usual.
Can you find what to invoke?
One one thing is neutral in the universe, that is $0$.
- Fm Jakaria
- Posts:79
- Joined:Thu Feb 28, 2013 11:49 pm
Re: Can you find what to invoke?
Any number of the form $4x+3$ works. This is because: $n^{\tau(n)}$ is a perfect square for all positive integers n.[If n is itself perfect square, done. Else $\tau(n)$ is even, and done.] And $x^2+y^2$ is 0 or 1 or 2 mod 4.
Note: I am not feeling good....
Note: I am not feeling good....
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.
Re: Can you find what to invoke?
" If the solution just crosses your mind, great!" did you miss that part? Because the latter part was kind of for newbies
One one thing is neutral in the universe, that is $0$.