Write the statement in symbolic language

Question

Consider the following statement: One plus the square of an odd natural number is divisible by a product of 2 consecutive natural numbers.

Write this in symbolic language. Is it true or false?

Answer 1

A natural number is odd iff it is of the form $2n+1$, with $n$ a natural number.

So we are considering numbers of the form $(2n+1)^2 +1 = 4n^2 + 4n + 2 = 2(2n^2 + 2n +1)$, but $10$ cannot be written as any product of consecutive numbers, so the statement is false. Two consecutive integers are of the form $m$ and $m+1$, so the statement can be written as

$$\forall n \exists m: (2n+1)^2 + 1= m(m+1)$$

where quantors are over the natural numbers.