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Number Theory If a³ ≡ b³ (mod n) then a ≡ b (mod n)

KOO

New member
Oct 19, 2013
19
Let $a, b \in Z$ and $n \in N$ . Is the following necessarily true?
If $a^3 ≡b^3$(mod n) then $a ≡ b$ (mod n)

I know it's false but I can't think of an counterexample.
 

kaliprasad

Well-known member
Mar 31, 2013
1,309
Re: If $a^3 ≡b^3$(mod n) then $a ≡ b$ (mod n)

Let $a, b \in Z$ and $n \in N$ . Is the following necessarily true?
If $a^3 ≡b^3$(mod n) then $a ≡ b$ (mod n)

I know it's false but I can't think of an counterexample.
No

$2^3 = 4^3$ mod 8
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,702
It can even happen with a prime modulus: $1^3 = 2^3\pmod7$.
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
Even in non-trivial cases : $4^3 = 10^3\pmod{13}$
 

johng

Well-known member
MHB Math Helper
Jan 25, 2013
236
Hi,
Here's an easy result in the positive direction.

If 3 is prime to \(\displaystyle \phi(n)\) and both a and b are prime to n, then the conclusion does follow.

Example: n=17 or any prime p with 3 not dividing p-1