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Chapter 9: Problem 37

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

Expert verified
There is one such solution for \( g \) fulfilling \( g^3 \equiv 2 \pmod{625} \).

Step by step solution

01

Parse the problem

We need to find the integer solutions \( g \) such that \( g^3 \equiv 2 \pmod{625} \). This means finding \( g \) in the range \( 0 \leq g < 625 \) where when raised to the power of 3 gives a remainder of 2 when divided by 625.
02

Factor the modulus

The modulus 625 can be written as powers of a single prime: \( 625 = 5^4 \). By analyzing these prime powers, we can separately solve smaller congruences modulo \( 5^4 \) and use them to derive a solution.
03

Find solutions modulo smaller powers of 5

First, solve \( g^3 \equiv 2 \pmod{5} \). We check integers in range 0 to 4 and find that \( g \equiv 3 \pmod{5} \). Next solve \( g^3 \equiv 2 \pmod{25} \) using solutions from step 2 and verify via substitution to find valid \( g \). Extend this to \( g^3 \equiv 2 \pmod{125} \) using similar checks.
04

Use Hensel's lemma

Using the solutions from step 3, we can extend to the higher power \( 625 = 5^4 \) using Hensel's lemma. This mathematical tool helps to lift solutions from a lower power of a modulus to a higher power if conditions like differentiability of the function are met. Extend \( g \equiv 3 \pmod{25} \) step by step to obtain \( g^3 \equiv 2 \pmod{125} \) and \( g^3 \equiv 2 \pmod{625} \) by checking small perturbations (small modifications around the garnished solution).
05

Verify final results

We verify by substituting possible \( g \) values back into the original equation \( g^3 \equiv 2 \pmod{625} \). If it satisfies the congruence, it is a valid solution. Check until solutions are exhausted within the range.
06

Count the number of solutions

At the end, tally how many valid \( g \) satisfy the constraints provided. Each particular solution lifted directly from smaller powers of the prime is considered to ensure correctness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Modular Arithmetic
Modular arithmetic is a mathematical system for dealing with integers where numbers "wrap around" after reaching a certain value, known as the modulus. This system can be visualized as the numbers on a clock face, where after reaching 12, the count starts again from 0 (or 1, depending on the system). In our exercise, the modulus is 625, and we're asked to find an integer cube root of 2 under this system.
  • When calculating something "modulo" a number, it means computing the remainder after division by that number.
  • For example, in our problem, we need to find an integer g such that when g is cubed, it has a remainder of 2 when divided by 625. This is expressed as the equation: \( g^3 \equiv 2 \, (\text{mod } 625) \)
  • Understanding this concept is pivotal as it helps reduce complex problems into manageable chunks.
Hensel's Lemma
Hensel's Lemma is a useful tool in number theory for finding solutions to polynomial equations modulo progressively higher powers of a prime. It's somewhat analogous to Newton's method for finding roots but within the modular arithmetic framework.
  • This lemma allows us to "lift" solutions of polynomials from lower moduli (like 5) to higher ones (like 25, 125, and 625 in our case).
  • Beginning with a known solution modulo a smaller power of a prime, Hensel's Lemma enables refining this solution step-by-step for larger powers, provided certain conditions hold, such as differentiability or the nature of the function involved.
  • In our original exercise, Hensel's Lemma helped lift the solution from modulo 25 up to modulo 125, and further to modulo 625.
Prime Factorization
Prime factorization involves breaking down a composite number into a product of prime numbers. This is essential for problems like those involving modular arithmetic because it often simplifies complex calculations.
  • In our exercise, we factorized 625 as \( 5^4 \) which made it straightforward to apply Hensel's Lemma.
  • By expressing the modulus (625) in terms of its prime factors, we can solve the original congruence by handling smaller components individually, and then utilize them to reassemble the complete solution.
  • This approach of breaking down into prime powers and solving is critical for systematically finding solutions in number theory.
Congruences
In mathematics, a congruence expresses that two numbers leave the same remainder when divided by a given modulus. It's a foundational concept in number theory with applications in cryptography, computer science, and more.
  • The original problem consists of the congruence equation \( g^3 \equiv 2 \, (\text{mod } 625) \).
  • Congruences are helpful when analyzing periodic processes or structures and simplify otherwise tricky problems by working in a "reduced" number system.
  • Using the basic properties of congruences, we determined potential solutions modulo smaller powers of 5, eventually solving \( g^3 \equiv 2 \) modulo 5, 25, 125, and 625.

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Most popular questions from this chapter

Let \(R\) be a ring (commutative, with 1 ), \(f \in R[x], u \in R\) a root of \(f\), and \(g=f /(x-u)\). Prove that \(f^{\prime}(u)=g(u)\).

Let \(R\) be a ring (commutative, with 1 ), \(f, g \in R[x]\), and \(n \in \mathbb{N}\). Prove that \(f \equiv g \bmod x^{n+1}\) implies \(f^{\prime} \equiv g^{\prime} \bmod x^{n}\), and give an example where \(f^{\prime} \equiv g^{\prime} \bmod x^{n+1}\) does not hold.

Let \(n \in \mathbb{N}, R\) be a ring such that \(n\) ! is a unit in \(R, f \in R[x]\) of degree \(n\), and \(f=\Sigma_{0 \leq i \leq n} f_{i} \cdot(x-u)^{i}\) its Taylor expansion around some \(u \in R\). Prove that \(f_{i}=f^{(i)}(u) / i !\) for all \(i\), where \(f^{(i)}\) is the \(i\) th derivative of \(f\).

Let \(a=30 x^{7}+31 x^{6}+32 x^{5}+33 x^{4}+34 x^{3}+35 x^{2}+36 x+37\) and \(b=17 x^{3}+18 x^{2}+19 x+20\) in \(\mathbb{F}_{101}[x]\), and \(f \in \mathbb{F}_{101}[x]\) the reversal of \(b\). (i) Compute \(f^{-1} \bmod x^{4}\). (ii) Use (i) to find \(q, r \in \mathbb{F}_{101}[x]\) with \(a=q b+r\) and \(\operatorname{deg} r<3\). (iii) Use the Extended Euclidean Algorithm to find \(a^{-1} \bmod b\), that is, a polynomial \(c \in F_{101}[x]\) of degree less than 3 with \(a c \equiv 1 \bmod b\). (iv) Use Newton iteration to find \(a^{-1} \bmod b^{4}\).

Find the Newton formula for approximating \(1 / \sqrt{a}\). What is the remarkable difference to the Newton formula for \(\sqrt{a}\) ?
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