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 October 21st, 2018, 06:07 AM #1 Senior Member     Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176 n=? 53|(5^n -1), n=?
 October 21st, 2018, 06:22 AM #2 Senior Member   Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry Have you heard of Fermat's little theorem? Thanks from topsquark
 October 21st, 2018, 07:20 AM #3 Senior Member     Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176 so...!?
 October 21st, 2018, 07:24 AM #4 Senior Member   Joined: Oct 2009 Posts: 609 Thanks: 186 Have you?
 October 21st, 2018, 07:54 AM #5 Senior Member     Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176 somewhat...
 October 21st, 2018, 08:10 AM #6 Global Moderator   Joined: Dec 2006 Posts: 19,868 Thanks: 1833 Have you considered n = 0?
 October 21st, 2018, 08:14 AM #7 Senior Member     Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176 n>0
 October 21st, 2018, 03:55 PM #8 Member   Joined: Oct 2018 From: Netherlands Posts: 39 Thanks: 3 n=52 appears to be a solution but there are probably other ones. Didn't find it by trying but not through solid proof either. The values for nx26 are promising candidates by virtue of Fermat's little theorem, but they need to be verified as not all do the job. Last edited by skipjack; October 21st, 2018 at 05:54 PM.
 October 21st, 2018, 06:19 PM #9 Global Moderator   Joined: Dec 2006 Posts: 19,868 Thanks: 1833 Any positive integer multiple of 52.
October 22nd, 2018, 03:47 AM   #10
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Joined: Aug 2017
From: United Kingdom

Posts: 282
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Math Focus: Algebraic Number Theory, Arithmetic Geometry
Quote:
 Originally Posted by skipjack Any positive integer multiple of 52.
Yes, that these numbers are solutions is immediate from Fermat's little theorem. The sightly more challenging thing is to show that there are no other solutions. But this problem can be reduced to showing that 4 and 26 are not solutions. Indeed, if n is a solution, then so is gcd(n,52) (by Bezout's lemma, for example). But if n is not a multiple of 52, then gcd(n,52) must divide 4 or 26 and so 4 or 26 would have to be a solution.

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