Factorial Sums

Number Theory Level 4

Find the sum of all positive integers \(m\) such that \(2^m\) can be expressed as sums of four factorials (of positive integers).

Details and assumptions

The number \( n!\), read as n factorial, is equal to the product of all positive integers less than or equal to \(n\). For example, \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

The factorials do not have to be distinct. For example, \(2^4=16\) counts, because it equals \(3!+3!+2!+2!\).


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