Quadratic Residues Modulo 3,4,5,7: Patterns & Analysis

[saadsarfraz]

Q- A number is a quadratic residue modulo m if it takes the form x^{2} mod m
for some integer x. List the quadratic residues modulo 3, 4, 5, and 7. What
patterns, if any, do you notice?

modulo 7

0^2=0, 1^{2}=1, 2^{2}=4, 3^{2}=2, 4^{2}=2, 5^{2}=4, 6^{2}=1

modulo 5

0^2=0, 1^{2}=1, 2^{2}=4, 3^{2}=4, 4^{2}=2

modulo 4

0^2=0, 1^{2}=1, 2^{2}=0, 3^{2}=1

modulo 3

0^2=0, 1^{2}=1, 2^{2}=1

I don't to see any patterns?

 

[CRGreathouse]

You forgot 0^2 = 0 in each case. But to see patterns, I think you're supposed to sort the residues and remove duplicates.

I'm not sure what pattern you're supposed to see, actually. That primes have more residues than composites? That about half the numbers are quadratic residues mod a prime? Or is it supposed to be related to the Law of Quadratic Reciprocity that you presumably haven't learned yet?

 

[saadsarfraz]

so if i remove duplicates then

0^2=0, 1^2=1, 2^2=4, 3^2=2 (mod7)

0^2=0, 1^2=1, 2^2=4, (mod5) i think 4^2=1 not 2.

0^2=0, 1^2=1 (mod4)

0^2=0, 1^2=1 (mod3)

 

[saadsarfraz]

what do you mean by "That about half the numbers are quadratic residues mod a prime?"