Show that 1 plus the product of the first n primes is never
a square, for any positive integer n.

Suppose that the product of the first n primes is a square for some
value of n.
Then we can write:

1 + 2.3.5...p_{n} = k^{2}

for some k, and so:

2.3.5...p_{n} = k^{2} - 1 = (k - 1)(k + 1)

Now, k is clearly odd, so (k - 1) and (k + 1) are both even.
Let k - 1 = 2r:

2.3.5...p_{n} = (2r)(2r+2)

3.5...p_{n} = 2r(r+1)

From this last equation, we can deduce that 2 must divide evenly
into some odd prime, which I hope you'll agree is absurd.
By contradiction, the result is established.