TITLE: A factorian is a number equal to the sum of the factorials of its digits, e.g., 145 = 1! + 4! + 5! and 40,585 = 4! + 0! + 5! + 8! + 5!. Let F be the largest factorian. Here I provide a very simple proof that F < 2,000,000.
AUTHOR: @games_to_change
#QUARK
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Step 2: F < 2,000,000. Proof: Suppose F > 2,000,000. Since F has 7 digits, F is at most 9! * 7 = 2,177,282. So, F’s 1st digit must be “2” and 2nd digit must be “1” or “0”. But then its factorial sum cannot exceed 2! + 1 + 9! * 5 = 1,814,403, less than 2,000,000, a contradiction!
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P.S. Interested readers can build on this argument to show that F < 1,500,000. Hint: F’s last digit cannot be “9” unless it contains a “1” or “0” and at least two more digits less than “5”.
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