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I’ve having difficulty understanding why 40585 is the greatest factorion that exists. Why can there not be one greater?
Wikipedia says, “This fails to hold for d ≥ 8.” In other words no factorion can exist with more than 7 digits.
But how is this known? How can it be proven? What about for very very large numbers, perhaps one that has not yet been tested?
For d=7 you have
There exist some n which fulfil this inequality, even though none of them actually is a factorion. For d=8 you get
As the left hand side is already larger than the right hand side, no value of n could possibly satisfy both inequalities at once. As the left hand side grows exponentially in d, and the right hand side only linearly, the problem will only become more dramatic, as the left hand side will grow much faster than the right hand side.
The reasons for the two bounds are simple: the number has to have at least d digits, and the smallest number with that many digits is 10d—1. On the other hand, it must be the sum of d factorials, each for a single digit, and the largest factorial you can obtain that way is 9!. Thus the inequality.