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:

rational rational.

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.

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