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October 21st, 2018, 06:07 AM   #1
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n=?

53|(5^n -1), n=?
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October 21st, 2018, 06:22 AM   #2
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Have you heard of Fermat's little theorem?
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October 21st, 2018, 07:20 AM   #3
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so...!?
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October 21st, 2018, 07:24 AM   #4
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Have you?
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October 21st, 2018, 07:54 AM   #5
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somewhat...
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October 21st, 2018, 08:10 AM   #6
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Have you considered n = 0?
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October 21st, 2018, 08:14 AM   #7
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n>0
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October 21st, 2018, 03:55 PM   #8
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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.
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October 21st, 2018, 06:19 PM   #9
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Any positive integer multiple of 52.
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October 22nd, 2018, 03:47 AM   #10
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Quote:
Originally Posted by skipjack View Post
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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