Solving Quadratic Congruences in Number Theory: A Comprehensive Guide

Quadratic congruences play a pivotal role in number theory, providing a rich framework for solving Diophantine equations. In this article, we will explore the methods to solve two specific quadratic congruences:

1. Solving the First Congruence: 7x2 - 9x - 4 ≡ 0 (mod 25)

We will break down the steps to find the solutions of the quadratic congruence (7x^2 - 9x - 4 equiv 0 pmod{25}).

Step 1: Calculate the Discriminant

The first step is to calculate the discriminant, which is given by D b2 - 4ac.

a  7b  -9c  -4D  (-9)2 - 4·7·(-4)  81   112  192

Next, we need to compute (192 mod 25):
192 ≡ 17 (mod 25)

Step 1.2: Find the Square Root of the Discriminant Modulo 25

We need to determine if 19 has a square root modulo 25 by checking squares modulo 25:

02 ≡ 0 (mod 25) 12 ≡ 1 (mod 25) 22 ≡ 4 (mod 25) 32 ≡ 9 (mod 25) 42 ≡ 16 (mod 25) 52 ≡ 0 (mod 25) 62 ≡ 6 (mod 25) 72 ≡ 24 (mod 25) 82 ≡ 24 (mod 25) 92 ≡ 6 (mod 25) 102 ≡ 0 (mod 25) 112 ≡ 16 (mod 25) 122 ≡ 9 (mod 25) 132 ≡ 4 (mod 25) 142 ≡ 1 (mod 25) 152 ≡ 0 (mod 25) 162 ≡ 16 (mod 25) 172 ≡ 9 (mod 25) 182 ≡ 4 (mod 25) 192 ≡ 1 (mod 25) 202 ≡ 0 (mod 25) 212 ≡ 24 (mod 25) 222 ≡ 24 (mod 25) 232 ≡ 6 (mod 25) 242 ≡ 24 (mod 25)

As we can see, 19 does not appear in the list, indicating that 19 does not have a square root modulo 25. Therefore, the original quadratic congruence (7x^2 - 9x - 4 equiv 0 pmod{25}) has no solutions.

2. Solving the Second Congruence: 7x2 - 41x 54 ≡ 0 (mod 75)

Moving on to the second quadratic congruence (7x^2 - 41x 54 equiv 0 pmod{75}).

Step 2.1: Calculate the Discriminant

We start by calculating the discriminant:

D  (-41)2 - 4·7·54  1681 - 1512  169

We then find the square root of 169 modulo 75:

sqrt{169} ≡ 13 (mod 75)

Step 2.2: Apply the Quadratic Formula

Substituting back into the quadratic formula:

x ≡ frac{-(-41) ± 13}{14} (mod 75)

This gives us two cases:

x? ≡ frac{54}{14} (mod 75)x? ≡ frac{28}{14} (mod 75)

For x?:

x? ≡ frac{54}{14} (mod 75)

We need to find the modular inverse of 14 modulo 75, which is 11, giving us:

x? ≡ 54 · 11 (mod 75)x? ≡ 594 (mod 75)

Simplifying, we get:

x? ≡ 72 (mod 75)

For x?:

x? ≡ frac{28}{14} (mod 75)

This simplifies to:

x? ≡ 2 (mod 75)

Therefore, the solutions to the original congruences are:

No solutions for the congruence (7x^2 - 9x - 4 equiv 0 pmod{25}). The solutions for the congruence (7x^2 - 41x 54 equiv 0 pmod{75}) are (x equiv 2 pmod{75}) and (x equiv 72 pmod{75}).

In conclusion, understanding and solving quadratic congruences involves several steps, including calculating the discriminant, finding the square root of the discriminant, and then using the quadratic formula. By following these steps, we can effectively solve complex number theory problems.