How to find an integer solution to a nonlinear equation?

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Discussion Overview

The discussion revolves around finding integer solutions to the nonlinear equation of the form \( a^n = c \), where \( a \), \( n \), and \( c \) are constrained to be integers. Participants explore various methods and approaches to identify values for \( a \) or \( n \) given a specific integer \( c \).

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that limiting the left-hand side to integers results in solutions only for specific choices of \( c \).
  • Another participant proposes finding the prime factors of \( c \) as a method to explore potential solutions.
  • It is mentioned that setting \( a = c \) and \( n = 1 \) provides a straightforward solution.
  • A participant illustrates a method involving prime factorization, providing an example with \( c = 1000 \) and discussing how to derive values for \( n \) that yield whole number exponents.
  • Another participant agrees with the prime factorization approach and presents a specific case where \( a = 10 \) and \( n = 3 \) for \( c = 1000 \).
  • One participant claims that the equation has an infinite number of solutions when \( c = 0 \) or \( c = \pm 1 \) and invites others to prove this formally.

Areas of Agreement / Disagreement

Participants express various methods for finding integer solutions, but there is no consensus on a single approach. Some methods are repeated, and while there is agreement on specific cases, the overall discussion remains unresolved regarding the generality of solutions.

Contextual Notes

Some methods rely on the factorization of \( c \), and there are assumptions about the nature of \( n \) and \( a \) being integers. The discussion does not resolve the formal proof of the claim regarding infinite solutions for specific values of \( c \).

Who May Find This Useful

This discussion may be useful for individuals interested in number theory, particularly those exploring integer solutions to nonlinear equations and the properties of exponents.

al4n
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given something like: an = c
where c is given and a, n, and c are only allowed to be integers. how would one find the value of say n or a?
 
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What have you found so far in your searching? It would seem that if you limit the LHS to integers, there are only solutions for specific choices of c, no?
 
The simplest approach is to find the prime factors of c.
 
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The easiest way is to just set ##a=c## and ##n=1##.
 
jedishrfu said:
The simplest approach is to find the prime factors of c.
thank you. this is very helpful. so I could write something like
an = 1000
= 2353
then like
a = 23/n53/n
and find values of n that result in whole number exponents. in this case 3, 1.
 
yes or you could look at ##a^n = 2^3 * 5^3 = (2 * 5)^3 ## and conclude a=10 and n=3

and of course the trivial case of a = 1000 and n=1
 
This equation also has the property that there is an infinite number of solutions only when ##c=0## or ##c=\pm 1## (can you prove this formally with mathematical induction or by some other way?).
 

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