Question

Compute a cube root of 2 modulo 625 , that is, \(g \in\\{0, \ldots, 624\\}\) such that \(g^{3} \equiv 2\) mod 625 . How many such \(g\) are there?

Short answer

There are no such \(g\) that satisfy \(g^3 \equiv 2 \pmod{625}\).

Step-by-step solution

Understanding the Problem

We are tasked with finding values of \(g\) such that \(g^3 \equiv 2 \pmod{625}\). This means that \(g\) raised to the third power, when divided by 625, gives a remainder of 2.

Finding the Prime Factorization of Modulus

We factor **625** to its prime components. Since \(625 = 5^4\), we will work with mod \(5^4\) to find the solutions.

Working modulo 5^4

Considering the equation \(g^3 \equiv 2 \pmod{625}\), we use the fact that since 625 is a power of a prime (5), we can potentially use Hensel's Lemma in the subsequent steps.

Finding g modulo 5

We start with checking possible values of \(g\) modulo 5. Possible values for \(g\) are 0, 1, 2, 3, 4.

Testing values by calculation

Calculate each: \(g^3 \equiv 2\) for each \(g\) mod 5. We find that \(g = 2\), since \(2^3 = 8 \equiv 3 \equiv 2 \pmod{5}\).

Lifting solutions to higher powers

Using Hensel's Lemma, lift the solution from 5 to 625. We find solutions for \(g\) in mod powers of 5, then test consistency in stepping up powers.

Final Computation and Solution Counting

Calculate \(g^3 = 2\mod 625\). Different occurrences for these values are tested for consistency and satisfaction of the condition: \(g^3 \equiv 2 \pmod{625}\).

Key concepts

Cube Root Modulo

When you're tasked with finding a cube root modulo some number, like in our case with 625, it means you are searching for a number, say \( g \), such that when you raise \( g \) to the power of three and divide it by the modulus, the remainder is specific. Here, we're interested in the equation \( g^3 \equiv 2 \, \text{mod} \, 625 \). This is a typical example of finding roots modulo a number, which requires both knowledge of modular arithmetic and deeper understanding of number properties, especially when the modulus is a power of a prime.

Cube roots modulo can be tricky, especially when working with composite numbers. In our exercise, this involves checking various possible values of \( g \) under smaller factors of the modulus to see which satisfy the equation. The process often involves trial and error and strategically narrowing down the field, which can be exhaustive but rewarding when finding the correct root.

This concept is particularly useful in number theory and cryptography, where knowing how to calculate and understand such roots is crucial.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. In other words, it's the expression of a number as a product of prime numbers. In our context, to solve the cube root modulo problem, we first factor the modulus. For the exercise, we factor 625.

Here's how it works:

• 625 is broken down into its prime factors.
• It turns into \( 5^4 \), meaning it's composed of the prime number 5 multiplied by itself four times.

Understanding this breakdown is essential because it allows us to apply specific mathematical tools like Hensel's Lemma more easily. It informs us how the modulus behaves under operations like division or multiplication of other numbers.

In solving polynomial congruences and related problems, knowing the prime factors of your modulus can greatly simplify the process, making your computations more manageable and insightful.

Hensel's Lemma

Hensel's Lemma is a powerful tool used in number theory to lift solutions of congruences from a prime modulus to higher powers of that prime. It is particularly useful when dealing with problems like finding roots modulo a number which is a power of a prime, just like 625, which is \( 5^4 \).

This lemma helps us to extend a solution found in a simpler context (modulo a smaller base, like 5) to a more complex one (modulo 625). Here’s the step-by-step application:

• First, solve the congruence modulo 5.
• Use Hensel’s Lemma to lift this solution to higher powers (like 25, then 125, and so on).

This lifting process involves "testing" solutions by checking their viability in each increasing power of 5. It's a systematic method that ensures the solutions fit the original congruence condition. This practical application of Hensel's Lemma is essential for advancing our initial findings in a feasible manner.

Polynomial Congruence

Polynomial congruence refers to equations like \( f(x) \equiv a \, \text{mod} \, m \) where the solutions involve polynomial expressions that are treated in modulo arithmetic fashion.

For the given exercise, the polynomial congruence we tackle is \( g^3 \equiv 2 \, \text{mod} \, 625 \). This asks if there can exist solutions of the polynomial \( g^3 - 2 \equiv 0 \, \text{mod} \, 625 \).

To solve this:

• Assess the polynomial in simpler modular forms first, like mod 5.
• Find roots that satisfy this simpler congruence.
• Use methods like Hensel’s Lemma to work upwards to the original modulus.

Understanding polynomial congruences is vital for applications in fields such as cryptography and coding theory. These applications rely on securely performing calculations over complex modular systems, which often involve solving these types of congruences with high precision and care.