Prove or Disprove: $a \equiv b \bmod m$ iff $a^3 \equiv b^3 \bmod m$?

Question

Prove or Disprove: $a \equiv b \mod m$ iff $a^3 \equiv b^3 \mod m$?

I'm trying to determine if this is true or not. I have already proved this going one way. I know that if $a \equiv b \mod m$ then $a^3 \equiv b^3 \mod m$.

How should I start the second direction? I know that I'm starting with $mk=b^3-a^3$ and I need to get down to $mj=b-a$ for some $k,j \in \mathbb{Z}$. Any hints on how to get there?

@Pawel That's essentially special case $,a,b = 0,2,$ in my initial comment. — Bill Dubuque, Sep 16 '20 at 22:55

score 1 · Accepted Answer · answered Sep 16 '20 at 23:05

1

It's easy to find lots of counterexamples. Take positive integers $b > a$ and look for numbers $m$ that divide $b^3 - a^3$ but don't divide $b-a$.

answered Sep 16 '20 at 23:05

Robert Israel

448,999

For the record: this hint was already given $30$ minutes prior in the first commen on the question. – Bill Dubuque Sep 17 '20 at 21:03

Prove or Disprove: $a \equiv b \bmod m$ iff $a^3 \equiv b^3 \bmod m$?

1 Answers1