Congruence Question

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Congruence Question !!

I have a question regarding congruences, I could not find this result in the textbooks.

(note to readers: a^k means a to the power k, and = means congruent)

If we have a congruence: a^m = a^n (mod p) for a,m,n,p>0

It seems likely to deduce that m = n (mod p)

However after attempting a homework question, I discover that

a^m = a^n (mod p) implies m = n (mod p-1)

Is this result true? How does one go about to formally prove the above statement?

Thank you...
 

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  • #2
shmoe
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What made you think that's true? Did you try some numerical examples?
 
  • #3
matt grime
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One would first go about thinking what the correct statement of the above should be: one raised to any power is still one.
 
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Consider a^m = a^n (mod p) => m = n (mod p) for m=p, n=1. Then it's obviously not true, and proof that it's not true follows immediately from the little fermat theorem.

Hint: the little fermat theorem is key to understanding the 2nd set of congruences as well
 
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