perplexus.info :: Numbers : Prime Test (3)

Prime Test (3) - Final (Posted on 2011-06-25)

Let P be any prime.

P is in the set S, unless P^72-1 is evenly divisible by 20174525280.

Using the results of Prime Tests (1) and (2), or some other method, prove that S is finite.

Optional: List the members of S.

Submitted by broll

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Solution: (Hide)

The problem is related to the Bernoulli Numbers, see A006863 in Sloane. Let P be any prime; then P^n-1 is divisible by A006863(n) unless (for obvious reasons)P itself is a factor of A006863(n).
20174525280 = 2^5×3^3×5×7×13×19×37×73, so those primes are all the members of S, and every other prime is so divisible.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Apologies	broll	2011-08-21 06:22:09

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