Prove or disprove: If x^2 p=1 has exactly two solutions x ∈{1,2, …, p-1}, then p is prime.

Name: Problem 6, Chapter 7: Applied Algebra: Codes, Ciphers and Discrete Algorithms
Uploaded: 2019-10-24T23:38:06-08:00
Duration: 1 min 59 s
Channel: James Chok
Description: Prove or disprove: If x^2 p=1 has exactly two solutions x ∈{1,2, …, p-1}, then p is prime.

step 1 Okay, so this question we want to prove that if x squared is equivalent to one mod p, and this i

Prove or disprove: If x^2 p=1 has exactly two solutions x...

Key Concepts

Quadratic Congruences

Quadratic congruences are equations of the form ax² + bx + c ? 0 (mod n) and their study includes analyzing the number and nature of solutions under different moduli. Understanding quadratic congruences is key to the problem in question, as the equation x² ? 1 (mod p) is a specific case where solution counts can reveal properties of the modulus, such as its primality.

Properties of Prime Numbers in Modular Systems

When the modulus is prime, the set of integers modulo that prime forms a field; this means every nonzero element has a multiplicative inverse and there are no zero divisors. This field property ensures that equations such as x² ? 1 (mod p) exhibit predictable behavior, typically having exactly two solutions (namely, x ? 1 and x ? -1), consistent with fundamental theorems in number theory.

Zero Divisors and Composite Moduli

In contrast to fields, the ring of integers modulo a composite number has zero divisors—nonzero elements whose product is zero modulo the composite modulus. This feature can lead to additional, nontrivial solutions in equations like x² ? 1 (mod n), because the absence of inverses and the existence of factorizations involving zero divisors complicate the structure of the solution set.

Modular Arithmetic

Modular arithmetic involves performing arithmetic operations within a set of integers modulo a given number, where numbers “wrap around” upon reaching this modulus. This concept is central to the study of congruences and allows one to discuss equations like x² ? 1 (mod p) in a well-defined manner.

Congruences and Modular Equations

A congruence is an equation that holds for integers under a modulus, meaning that two numbers differ by a multiple of that modulus. Solving modular equations, like quadratic congruences, often involves techniques such as factorization and reduction, providing insights into the structure and solutions of equations in modular systems.

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