Problem 4

Find all pairs

(k,n)

of positive integers such that

k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}).

Step 5 of 6: Check the five remaining values of n

Previous step Next step

L_3=168,\qquad L_4=20160,\qquad L_5=9999360,\qquad 5!<L_3<6!,\quad 7!<L_4<8!,\quad 10!<L_5<11!

Detailed analysis

It remains to check $n=1,2,3,4,5$ . The first two values were handled in Step 1. For the others, $L_3=168$ lies strictly between $5!$ and $6!$ , $L_4=20160$ lies strictly between $7!$ and $8!$ , and $L_5=9999360$ lies strictly between $10!$ and $11!$ . Since factorials are strictly increasing, none of these three products is a factorial.