I’ve been working on an interesting task from a regional math contest:
Prove that the product of 8 successive naturals cannot be a natural number to the power of 4.
To prove this, we will first take a look at two other theorems (and prove them), and then use them to prove the original statement.
I. irrational irrational
To prove this, it suffices proving the contrapositive:
We have that for some a, b,
Square both sides to get . Thus, is rational.
II. is irrational
To prove this, note that x and (x+1) need to be squares. Consider for some a, . Further, for some b, .
Now, . But the only way this is possible if .
Since a and b are positive naturals, we reach a contradiction for the identity above and thus either x or x+1 have no squares. In either case, is irrational.
III. Prove that there is no y s.t.
We will assume that such y exists and reach a contradiction.
Rewrite as and suppose y is rational.
From I we have that it suffices to only prove that is rational.
From II we have that either x or (x+1) is irrational. At least one of the 8 elements has no square, and we reach a contradiction. Thus y is irrational.