Cubic residue

modulo $m$

An integer $a$ for which the congruence $x^3=a$ ($\bmod\,m$) is solvable. If the congruence has no solution, $a$ is called a cubic non-residue modulo $m$. If the modulus is a prime number $p$, the congruence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using Euler's criterion: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solvable if and only if

$$a^{(p-1)/q}\equiv1\pmod p,$$

where $q=(3,p-1)$. When the condition is satisfied, the congruence has exactly $q$ distinct solutions modulo $p$. It follows from the criterion, in particular, that for a prime number $p$, the sequence of numbers $1,\dots,p-1$ contains exactly $(q-1)(p-1)/q$ cubic non-residues and $(p-1)/q$ cubic residues modulo $p$.

Comments

From class field theory one obtains, e.g., that $2$ is a cubic residue modulo a prime number $p \equiv 1 \pmod 3$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).

See also Quadratic residue; Power residue; Reciprocity laws; Complete system of residues; Reduced system of residues.

References

• Cox, David A. Primes of the form $x^2+n y^2$. John Wiley & Sons (1989) ISBN 0-471-50654-0. Zbl 0701.11001