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- April 3rd 2011, 05:58 AM #1

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- April 3rd 2011, 07:25 AM #2

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- April 3rd 2011, 08:50 AM #3

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- April 3rd 2011, 08:52 AM #4

- April 3rd 2011, 09:06 AM #5

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- April 3rd 2011, 09:24 AM #6

- April 3rd 2011, 10:04 AM #7

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i feel i am missing something, too. 5 has order 4 in U(Z26), so i don't see how listing the values for n (mod 3) can possibly be right, whereas 3 has order 3. furthermore, 5^n (mod 26) never takes on any of the values 9,11 or 17 (mod 26). i don't see ANY solutions. and i don't agree with tonio's values of 5^n at ALL.

- April 3rd 2011, 10:50 AM #8
I can show that n, if it exists, must be an even number...Briefly, consider

Divide both sides by 2 and we see that the remaining factor on the LHS must have an even number of terms. Thus n - 1 is odd and n is even.

I can't seem to make it beyond this point.

-Dan

Edit: I might be able to finish this (proving there are no solutions), but I'm not certain on the logic. I'll throw it out.

If n is even let it be n = 2a. Then

But in there are no even factors of 1. Thus a doesn't exist, and nor does n.

- April 3rd 2011, 03:17 PM #9

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- April 3rd 2011, 06:06 PM #10

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divide by 2? 2 is a zero-divisor in Z26, i don't think that's a good idea.....

5^n = 5 ≠ 11 = 3^n + 8 if n ≡ 1 (mod 12)

5^n = 25 ≠ 17 = 3^n + 8 if n ≡ 2 (mod 12)

5^n = 21 ≠9 = 3^n + 8 if n ≡ 3 (mod 12)

5^n = 1 ≠ 11 = 3^n + 8 if n ≡ 4 (mod 12)

5^n = 5 ≠ 17 = 3^n + 8 if n ≡ 5 (mod 12)

5^n = 25 ≠ 9 = 3^n + 8 if n ≡ 6 (mod 12)

5^n = 21 ≠11 = 3^n + 8 if n ≡ 7 (mod 12)

5^n = 1 ≠ 17 = 3^n + 8 if n ≡ 8 (mod 12)

5^n = 5 ≠ 9 = 3^n + 8 if n ≡ 9 (mod 12)

5^n = 25 ≠ 11 = 3^n + 8 if n ≡ 10 (mod 12)

5^n = 21 ≠17 = 3^n + 8 if n ≡ 11 (mod 12)

5^n = 1 ≠ 9 = 3^n + 8 if n ≡ 0 (mod 12)

no solution.....

- April 3rd 2011, 07:47 PM #11