Residue Classes

Consider the diophantine equation -

$x^3 + y^4 = 7$

The solution which was given in the book had argument starting like - "Consider the residue class of $x^3$ modulo 13"

From where does one get motivation to check the residue class of that particular modulo?

Easy ones can be seen directly like checking residue class of modulo 3 in case of squares, modulo 7 in case of cubes, but what about others?

And yes, is there any list available of frequently used residue classes of modulo x? I think that might help many students. :)

#NumberTheory #Advice #Math

Easy Math Editor

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Comments

You are looking for prime numbers that have a lot of cube roots and fourth roots of unity. Modulo $13$ , there are $3$ cube roots of unity ( $1,3,9$ ) and $4$ fourth roots of unity ( $1,5,8,12$ ), and so there are exactly $4$ cubes and $3$ fourth powers modulo $13$ . This cuts down the number of possible residues of $x^3+y^4$ down to size (and, specifically, misses $7$ , even if that is the only residue that gets missed out).

Mark Hennings - 7 years, 6 months ago

Great explanation! To expand slightly, the primitive element theorem tells us that modulo $p$ there is a multiplicative generator $g$ , such that the order of $g$ is $p-1$ . Hence, we would have 3 cube roots if and only if $p-1$ is a multiple of 3. (Otherwise, there is only 1 cube root, namely 1.) In turn, this implies that there are $\frac{p-1}{3}$ possible values of cubes (and otherwise, there are $p-1$ values).

And if 13 didn't work, what would be the next number we try, given the same motivation?

Calvin Lin Staff - 7 years, 6 months ago

$37$ , being the next prime of the form $12n+1$ .

Mark Hennings - 7 years, 6 months ago

I think for cubes always check modulo 7,9, and 13 because the cubes leave very less numbers of distinct remainders modulo these numbers.

Kishan k - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$