A Curious Connection between Fermat's Number and Multiple Factoriangular Numbers

In the seventeenth century Fermat defined a sequence of numbers Fn=22n +1 for n ? 0 known as Fermat?s number. If Fn happens to be prime then Fn is called Fermat prime. All the Fermat?s numbers are of the form n!k+ ?nk for some fixed value of k and n. Further we will prove that after F4 no other Fermat prime exist upto 1050 .