Question

Show that if \(p\) is prime and \(p \geq 7\), then there are always two consecutive quadratic residues of \(p\). Hint: Show that at least one of 2,5 or 10 is a quadratic residue of \(p\).

Short answer

Question: Show that if p is a prime number greater than or equal to 7, there will always be two consecutive quadratic residues of p. Answer: We proved that there exist two consecutive quadratic residues of p for a prime number p ≥ 7 by using the hint that at least one of the numbers (2, 5, or 10) is a quadratic residue of p. In each case, we were able to show the existence of two consecutive quadratic residues using the properties and behavior of quadratic residues.

Step-by-step solution

Understanding Quadratic Residues

A quadratic residue (mod p) is a number x such that there exists an integer y with \(x \equiv y^2\pmod{p}\), where p is a prime number. We know from the exercise that \(p\) must be a prime number greater than or equal to \(7\). We are given a hint that at least one of the numbers 2, 5, or 10 is a quadratic residue of \(p\). To show that there exist two consecutive quadratic residues for \(p\), we need to examine the quadratic residue properties of 2, 5, and 10 (mod p).

Exploring the Behavior of Quadratic Residues

Let's consider the following congruences (mod p): 1. \((a - 1)^2 \equiv a^2 - 2a + 1 \pmod{p}\) 2. \(a^2 \equiv a^2 \pmod{p}\) 3. \((a + 1)^2 \equiv a^2 + 2a + 1 \pmod{p}\) There are two important deductions we can make here. If either a or b is a quadratic residue, then so is their product. Secondly, as a ranges from 0 to (p-1)/2, we can see that the values of a^2 (mod p) are distinct.

Confirming the Hint

Using these findings, we can confirm that at least one of the numbers (2, 5, or 10) is a quadratic residue of p. We know that since p is an odd prime number, either 2 or \(-1 = p - 1\) is a quadratic residue. Using a similar argument, either 5 or \(-1 = p - 1 \pmod{p}\) is a quadratic residue. Thus, at least one of the three numbers (2, 5, 10) is a quadratic residue of \(p\).

Proving the Existence of Consecutive Quadratic Residues

Now that we have verified the hint, let's use it to prove that there exist two consecutive quadratic residues for a prime number \(p \geq 7\). If 2 is a quadratic residue, we can conclude that there are at least two consecutive quadratic residues in the form \((a - 1)^2, a^2 \pmod{p}\). Similarly, if 5 is a quadratic residue, we can find consecutive quadratic residues in the form \(a^2, (a + 1)^2 \pmod{p}\). Lastly, if 10 is a quadratic residue, we can use consecutive quadratic residues in the form \((a - 1)^2, a^2 \pmod{p}\). In each case, we were able to show the existence of two consecutive quadratic residues using the hint that at least one of the numbers (2, 5, or 10) is a quadratic residue of \(p\). Therefore, we have shown that if a prime number \(p \geq 7\), there will always be two consecutive quadratic residues of \(p\).

Key concepts

Number Theory

When diving into the world of mathematics, number theory emerges as a fundamental branch that deals specifically with the properties and relationships of integers. It is a vast field with many subtopics, one of which involves quadratic residues, a concept central to our problem. In the realm of number theory, understanding quadratic residues is critical for solving various types of congruences and other mathematical puzzles.

Number theory has a practical application in modern-day cryptography, especially when dealing with prime numbers and residues. Deciphering the complex relationships between numbers allows us to create secure electronic transactions, protecting sensitive information from undesired access. As such, the study of number theory isn't just an abstract pursuit; it has real-world implications that we encounter in our digital lives.

Prime Numbers

At the heart of our exercise lies the concept of prime numbers. These unique numbers are greater than 1 and can only be divided by 1 and themselves without leaving a remainder. They're the building blocks of all natural numbers, making them incredibly important in number theory. In our case, we're taking a prime number, denoted by p, and examining its behavior in congruences concerning quadratic residues.

Unique Properties of Prime Numbers

One of the interesting aspects is that when a number is a prime, its divisibility rules are less complex, which often simplifies the analysis of congruences. The problem hinges on the fact that a prime number has characteristics that will always result in at least one of the numbers 2, 5, or 10 being a quadratic residue when considering congruences modulo p. These unique properties help us to navigate through complex number theory problems with more ease and understanding.

Congruences

The exercise places significant emphasis on the idea of congruences, a term frequently mentioned in number theory. Congruences are a way of expressing that two numbers leave the same remainder when divided by a third number, termed the modulus. In our case, the modulus is the prime number p.

Congruences appear frequently in cryptology, computer science, and modular arithmetic puzzles. Specifically, a quadratic residue modulo p represents a number that, when squared, is congruent to another number modulo p. Understanding congruences allows us to solve equations within a specific system of modular arithmetic, as we did in our textbook exercise.

The concept helps elucidate the behavior and properties of numbers as they relate to one another within a prescribed set of rules. This deeper understanding is crucial for resolving the existence of two consecutive quadratic residues for a given prime number, as shown in the textbook solution.