Math Problem Statement

In this question, we will find all integer solutions to the Diophantine equation x^2 = y^5. Let's first consider x, y >= 2. (a) By the FTA, there are primes p1 < p2 < ... < pk and q1 < q2 < ... < qe and ri, sj in N so that x = p1^r1 * p2^r2 * ... * pk^rk and y = q1^s1 * q2^s2 * ... * qe^sq. Plugging this into x^2 = y^5, we get p1^(2r1) * p2^(2r2) * ... * pk^(2rk) = q1^(5s1) * q2^(5s2) * ... * qe^(5sq). Complete the following statement: By the Fundamental Theorem of Arithmetic, these two factorizations are equal so k = e, p_i = q_i, and 2r_i = 5s_i for 1 <= i <= k. (b) In Example 11.12 of the notes, we showed that all integer solutions to the equation 2x = 5y are all pairs of integers x, y in the set {(3u, 2u): u in Z}. Thus, for some u in Z, x = 3u and y = 2u. If we let z = p1^r1 * p2^r2 * ... * pk^rk, find an expression for all integer solutions (x, y) of the Diophantine equation x^2 = y^5 in terms of z, when x, y >= 2.

Solution

The image you uploaded contains a mathematical quiz question from an assessed quiz. The problem is asking for integer solutions to the Diophantine equation x2=y5x^2 = y^5, and is connected to the Fundamental Theorem of Arithmetic and the Factorization Theorem. Here's a breakdown of what is being asked:

  • Part (a) refers to prime factorizations of x2x^2 and y5y^5. It's asking for the factorization of these terms in terms of primes.
  • Part (b) discusses solutions to the equation x2=y5x^2 = y^5 for integer pairs. The question provides a relationship between integer solutions and asks to derive the general form of solutions.
  • Part (c) seems to ask for filling in specific information related to integer solutions of x2=y5x^2 = y^5.

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Math Problem Analysis

Mathematical Concepts

Diophantine Equations
Prime Factorization
Fundamental Theorem of Arithmetic

Formulas

Prime factorization of x^2 and y^5
2r_i = 5s_i

Theorems

Fundamental Theorem of Arithmetic

Suitable Grade Level

University-level mathematics (Number Theory)