I am completely puzzled at the following question, so please if able write the steps with explanation, thank you in davance Let Р be a prime. An integer 0 ≤ q < p is called a quadratic residuemodulo P if there exists an ng € Z with q = n² (mod p). These were studied by suchfamous mathematicians as CARL FRIEDRICH GAUSS (1777-1855) and ADRIEN-MARIELEGENDRE (1752-1833).Follow in their footsteps by showing that there are precisely (p+1) quadratic residuesmodulo p whenever p is an odd prime. What about p = 2?

Answered: Follow in their footsteps by showing…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I am completely puzzled at the following question, so please if able write the steps with explanation, thank you in davance

Let Р be a prime. An integer 0 ≤ q < p is called a quadratic residue
modulo P if there exists an ng € Z with q = n² (mod p). These were studied by such
famous mathematicians as CARL FRIEDRICH GAUSS (1777-1855) and ADRIEN-MARIE
LEGENDRE (1752-1833).
Follow in their footsteps by showing that there are precisely (p+1) quadratic residues
modulo p whenever p is an odd prime. What about p = 2?

Expert Solution

Step 1: Step 1

Consider the set of integers from 1 to p-1 (inclusive). This set contains p-1 integers.

Since p is an odd prime, we can pair up the residues into sets of two. For each quadratic residue x, there is a corresponding residue -x (which is also a quadratic residue since $left parenthesis negative y right parenthesis squared equals y squared$ ).