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

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In summary, a number is a quadratic residue modulo m if it takes the form x^{2} mod m for some integer x. The quadratic residues for modulos 7, 5, 4, and 3 are 0, 1, 2, and 4. When removing duplicates, we get 0, 1, 2 (mod7), 0, 1, 4 (mod5), 0, 1 (mod4), and 0, 1 (mod3). There are no clear patterns in the residues, but it can be noted that about half of the numbers are quadratic residues modulo a prime number.
  • #1
saadsarfraz
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Q- A number is a quadratic residue modulo m if it takes the form x[tex]^{2}[/tex] 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[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=4, 3[tex]^{2}[/tex]=2, 4[tex]^{2}[/tex]=2, 5[tex]^{2}[/tex]=4, 6[tex]^{2}[/tex]=1

modulo 5

0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=4, 3[tex]^{2}[/tex]=4, 4[tex]^{2}[/tex]=2

modulo 4

0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=0, 3[tex]^{2}[/tex]=1

modulo 3

0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=1

I don't to see any patterns?
 
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  • #2
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?
 
  • #3
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)
 
  • #4
what do you mean by "That about half the numbers are quadratic residues mod a prime?"
 

1. What are quadratic residues modulo n?

Quadratic residues modulo n are the set of numbers that, when squared and divided by n, leave a remainder of 1. These numbers have special properties and can be used in various mathematical applications.

2. How are quadratic residues calculated?

To calculate the quadratic residues modulo n, we first find the square of all numbers from 1 to n. Then, we divide each square by n and take the remainder. If the remainder is 1, then the number is a quadratic residue modulo n.

3. What is the significance of studying quadratic residues modulo n?

Studying quadratic residues modulo n can help us understand number patterns and properties, such as divisibility, prime numbers, and cryptography. It also has applications in fields like number theory, algebra, and computer science.

4. What are the patterns of quadratic residues modulo 3, 4, 5, and 7?

The patterns of quadratic residues modulo 3, 4, 5, and 7 are as follows:

  • Modulo 3: The quadratic residues are 1 and 4.
  • Modulo 4: The quadratic residues are 1 and 0.
  • Modulo 5: The quadratic residues are 1 and 4.
  • Modulo 7: The quadratic residues are 1, 2, and 4.

5. How can quadratic residues modulo n be used in cryptography?

Quadratic residues modulo n have been used in various cryptographic schemes, such as the RSA algorithm, to ensure the security of data transmission. By choosing appropriate values for n and the quadratic residue, we can create a one-way function that is difficult to reverse without the correct decryption key.

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