For some reason, which I don't quite understand, ALL the Prime Factors of the product[1! 2! 3!...100!] that have an ODD exponent become EVEN exponents when divided by 50! as can be seen in the following breakdown:
[(6941 digits) = 2^4731 × 3^2328 × 5^1124 × 7^734 × 11^414 × 13^343 × 17^250 × 19^220 × 23^174 × 29^129 × 31^117 × 37^91 × 41^79 × 43^73 × 47^61 × 53^48 × 59^42 × 61^40 × 67^34 × 71^30 × 73^28 × 79^22 × 83^18 × 89^12 × 97^4] /
[50! = 2^47 * 3^22 * 5^12 * 7^8 * 11^4 * 13^3 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47]
= (6876 digits) =[ 2^4684 × 3^2306 × 5^1112 × 7^726 × 11^410 × 13^340 × 17^248 × 19^218 × 23^172 × 29^128 × 31^116 × 37^90 × 41^78 × 43^72 × 47^60 × 53^48 × 59^42 × 61^40 × 67^34 × 71^30 × 73^28 × 79^22 × 83^18 × 89^12 × 97^4].
Since ALL the Prime Factors, after division by 50!, have EVEN exponents, it therefore follows that the final result is a perfect square! Is it because all the Prime Factors of 50! are repeated in the remaining 51!....100! ?? I don't know.
