How to prove the product of first n consecutive odd numbers is a square less than another square?

Question

I have observed for first few values of consecutive odd numbers, the result is always of the form:

$m^2 - n^2$, where $m$ and $n$ are two distinct positive integers. That is: $$1\cdot 3\cdot 5 \cdot 7\cdots (2k-1) = m^2 - n^2$$

For example:

$ 1\cdot 3 = 3 = 2^2 - 1^2 \\ 1\cdot 3\cdot 5 = 15 = 4^2 - 1^2 \\ 1\cdot 3\cdot 5\cdot 7 = 105 = 11^2 - 4^2 \\ 1\cdot 3\cdot 5\cdot 7\cdot 9 = 945 = 31^2 - 4^2 \\ \vdots $

But not sure, how to prove it. Here is an attempt using induction:

Let it be true for some value of k, that is: $1\cdot 3\cdot 5\cdot 7\cdot 9\cdots(2k - 1) = m^2 - n^2$

Then when $k$ takes the value of $k+1$, we have

$$\begin{align} 1\cdot 3\cdot 5\cdot 7\cdot 9\cdots(2k - 1)\cdot(2k + 1) &= (m^2 - n^2)\cdot(2k + 1)\\ &= (m^2 - n^2)\cdot {(k+1)^2 - k^2} \end{align}$$ and got stuck here.

Can you please suggest to proceed further or an altogether a different way of proving so or prove me wrong.

Thanks in advance.

Answer 1

Note that any product of two distinct odd numbers or two distinct even numbers can be written as a difference of squares. Let $p<q$ be both odd or both even. Then $(q-p)/2$ and $(q+p)/2$ are distinct integers, and $$ ((q+p)/2)^2 - ((q-p)/2)^2=(q^2/4+pq/2+p^2/4)-(q^2/4-pq/2+p^2/4)=pq. $$ So this is a fairly weak condition, and one that the product of consecutive odd numbers certainly meets.

Answer 2

You might note that the difference sequence of the sequence of squares is exactly the odd numbers. Here's the sequence of squares:

$$0, 1, 4, 9, 16, 25, 36, \ldots$$

Here's the sequence of the differences:

$$1, 3, 5, 7, 9, 11, \ldots$$

That means every odd number is the difference of two consecutive squares. And your number is odd.