MATH437 December 2011
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
• Q1 • Q2 • Q3 • Q4 • Q5 • Q6 • Q7 (i) • Q7 (ii) •
Let such that . Prove that
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
If m and n differ, then there is a largest prime which they differ. Starting with that, argue using a descent argument.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
The claim is clearly true of one of m or n equals 1. Thus we suppose that both are strictly greater than 1.
In this case, the Fundamental Theorem of Arithmetic tells us that we may write
where each of the exponents is a positive integer at least 1. The left hand side of the equation in question thus reduces to
and in total as we are given that , we have
Now, suppose that m and n are distinct. Let be the largest prime where they differ and without loss of generality, suppose that but . Then we know that we must have that for some . Thus .
However, was chosen to be the largest such prime where m and n differ to any power. So for some and thus . Thus, we may cancel out these terms and repeat this argument to find another index and since there are only finitely many of these terms, we must eventually reach a contradiction.
Hence m and n have all the same prime factors to the same powers and thus are equal.
Food for thought: Is there an easier proof of this using the fact that
where the above sum ranges over all integers with ?
Click here for similar questions
MER QGQ flag, MER RH flag, MER RS flag, MER RT flag, MER Tag Divisibility, MER Tag Fundamental theorem of arithmetic, MER Tag Multiplicative functions, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag