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...pn = k2
for some k, and so:
2.3.5...pn = k2 - 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...pn = (2r)(2r+2)
3.5...pn = 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.