Congruence of all integers n, 4^n and 1 +3n mod(9)?

  • Thread starter mgiddy911
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  • #1
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I just took a number theory midterm, the professor had a question the that said
"Show by induction that for all integers n, 4[tex]^{n}[/tex] is congruent to 1 +3n mod(9).

Now am I crazy or did the professor probably mean to say integers greater or equal to 0, or for any natural number n, ...

couldn't you show a counter example for instance n = -2, such that the congruence is false?
 

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  • #2
CRGreathouse
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I just took a number theory midterm, the professor had a question the that said
"Show by induction that for all integers n, 4[tex]^{n}[/tex] is congruent to 1 +3n mod(9).

Now am I crazy or did the professor probably mean to say integers greater or equal to 0, or for any natural number n, ...

couldn't you show a counter example for instance n = -2, such that the congruence is false?

In number theory, it's not unusual to assume that variables are natural numbers unless otherwise specified. I don't really want to think about negative powers here, since:
* There is no multiplicative inverse over natural numbers or integers
* There is a multiplicative inverse over real numbers (x not 0)
* There is sometimes a multiplicative inverse over Zp

and there could be reason to work over any of these.
 
  • #3
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The only residues are 1,4,7, so even if we use the inverses, it doesn't matter since

[tex]\frac{1}{4}\equiv 7 mod 9[/tex]
 
  • #4
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When you think about it, as a general rule, if a and Mod b are such that, a, b positive integers, and (a,b) =1, (they are relatively prime ) Then the powers of a Mod b form a cyclic group, i.e., every power of a has an inverse in the group.
 
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