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April 8th, 2014, 03:11 AM   #1
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Induction again

Prove that (4^n) - 1 is divisible by 3

My attempt:

1) let n = 1
(4^1) - 1 = 4 - 1 = 3
3 is divisible by 3. So true for n = 1

2) Assume it is true for n = k
So (4^k) - 1 is divisible by 3

3) let n = k + 1


but and are divisible by 3

so is divisible by 3

Therefore for all n 4^n - 1 is divisible by 3

Is this proof acceptable? The third step doesn't look like induction..thanks
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April 8th, 2014, 03:19 AM   #2
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Quote:
Originally Posted by shunya View Post
3) let n = k + 1


but and are divisible by 3
$4^k+1$ should be replaces by $4^k-1$ (two times). Otherwise, the proof is correct. For clarity, it would be nice to say that $3\mid(4^k-1)$ by induction hypothesis.
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