No primorial plus one has ever been found that has square factors. However, when I started looking at the square of primorials plus one, I found:
(23#)^2 + 1 = 29^2 * 53 * 1116604864937
I've only searched through 107# so far, but I haven't found any other examples of this happening. Has anyone researched this extensively for powers of primorials and/or have a compelling argument for or against any primorial plus one having a square factor? Thanks for your help :).
PS: The primes (below 107#) are the squares plus one of 2#, 3#, 7#, 11#, 19#, and 53#.