Algebra
Draft for Information Only ContentDiophantine Equation Diophantine EquationSource: An Introduction to Diophantine Equations  A ProblemBased Approach A Diophantine equation is a polynomial equation with two or more unknowns of which only integer solutions are interested. In other words, a Diophantine equation is an equation of the form, ƒ(x₁,x₂,…,xₙ)=0 where ƒ is an nvariable function with n≥2. A Diophantine equation is an algeraic Diophantine equation, if ƒ is a polynomial with integral coefficients. An nuple (x₁⁰,x₂⁰,…,xₙ⁰)∈ℤⁿ satisfying the algebraic Diophantine equation is called a solution to the algebraic Diophantine equation. An equation having one or more solutions is called solvable. The most common general bivariate quadratic Diophantine equation is of the form, Ax² + Bxy + Cy² + Dx + Ey + F = 0 in which all variables and solutions should be integer numbers in general. The three basic problems of a Diophantine equation are
Solving Diophantine EquationThe Factoring Method
For a given Diophantine equation ƒ(x₁,x₂,…,xₙ)=0, if the equation can be expressed in the equivalent form
The complete set of solutions to the Diophantine equation can be obtained by solving the system of equations. For example
x²y²=pq Method of IntervalFor a given Diophantine equation ƒ(x₁,x₂,…,xₙ)=0, if the variables of equation can be restricted within intervals by appropriate inequalities, then the solutions of the equation may lead to only finitely many possibile values. For example x³+y³=(x+y)²,
Because of square terms on the left hand side and the sum of 2 on the right hand side, one of the three terms on the left hand side must be equal to zero and each term on the left hand side must be less than or equal to 1, (xy)₂≤1, (x1)₂≤1, and (y1)₂≤1. Therefore the variables x, y are restricted between the interval [0,2]. The other solutions of the equation is then equal to (0,1), (1,0), (1,2), (2,1), and (2,2) besides (k,k). The Parametric MethodIn some cases, the integral solutions of a Diophantine equation, ƒ(x₁,x₂,…,xₙ)=0, can be expressed in a parametric form, that is x₁=g₁(k₁,k₂,…,kₗ), x₂=g₂(k₁,k₂,…,kₗ), …, x₂=gₖ(k₁,k₂,…,kₗ), where g₁, g₂, …, gₖ are integervalued lvariable functions and k₁,k₂,…,kₗ∈ℤ. Sometimes the set of solutions may have multiple parametric representations. However, for most Diophantine equations it is not possible to find all solutions explicitly. In many such cases,the parametric method provides a proof of the existence of infinitely many solutions. For example x³+y³+z³=x²+y²+z² The Modular Arithmetic MethodIn some cases, simple modular arithmetic considerations are used to prove the Diophantine equation is not solvable or in reducing the range of their possible solutions. For example(x+1)²+(x+2)²+…+(x+2001)²=y² The Method of Mathematical InductionMathematical induction is a tool for proving statements depending on nonnegative integers. Let (P(n))n≥0 be a sequence of propositions. The method of mathematical induction can then be used to prove that P(n) is true for all n≥n₀, where n₀ is a given nonnegative integer. Mathematical Induction (weak form)Suppose
Then P(n) is true for all n≥n₀. Mathematical Induction (with step s)Let s be a fixed positive integer. Suppose
Then P(n) is true for all n≥n₀. Mathematical Induction (strong form)Suppose
Then P(n) is true for all n≥n₀. For examplex²+y²+z²=59ⁿ, there exist positive integers x,y, z for all integer n≥3 Let step s=2 and n₀=1 For n≥3, let xₙ₊₂=59xₙ, yₙ₊₂=59yₙ, zₙ₊₂=59zₙ Therefore, for all n≥1 then In other words, for all n≥1 then Fermat's Method of Infinite Descent (FMID)The method of infinite descent is an indirect method that is proved by contradiction and relies on the least integer principle. One typical application is to prove an equation has no solutions or is false for all large enough nonnegative integer. Both finite and infinite descent methods assume a property P concerning the nonnegative integers and a nonnegative integer n such that (P(n))n≥1 be the sequence of propositions where {P(n): n satisfies property P}. To prove the proposition P(n) is false for all large enough n.
Then P(n) is false for all n≥k. That is, P(n) is false for any nonnegative integer n since P(k) is not true. For the Fermat's Method of Infinite Descent (FMID), let k be a nonnegative integer. Suppose that:
Then P(n) is false for all n>k. That is, P(n) is false for any nonnegative integer n since no infinite descending sequence, P(n) exists for nonnegative integers n, such that n₁>n₂>⋯>k. In other words, there is no infinite descending sequence of nonnegative integers, n₁>n₂>⋯. if P(k) is not true, then there is no no infinite descending sequence of nonnegative integers, n₁>n₂>⋯>k and If n₀ is the smallest positive integer n for which P(n₀) is true, then P(n) is false for all n<n₀. And, If the sequence of nonnegative integers (nᵢ)i≥1 satisfies the inequalities n₁≥n₂≥⋯ then there esists i₀ such that nᵢ₀=nᵢ₀₊₁=⋯. For examplex³+2y³=4z³, (0,0,0) is the only solution and there are no other nonnegative integer solutions. Assume (x₁, y₁, z₁) is a nontrivial solution. Since ³√2, ³√4 are both irrational, x₁>0, y₁>0, z₁>0. Subsitute (x₁, y₁, z₁) into x³+2y³=4z³, imply x₁³+2y₁³=4z₁³,
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