Science:Math Exam Resources/Courses/MATH312/December 2008/Question 04 (c)
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Question 04 (c) 

Show, without using the explicit prime factorization of 1729, but using the following congruences instead, that 1729 is composite.

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! 
Hint 

This idea is similar to the idea in Miller's Test. Suppose that 1729 is prime. How many solutions should have? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Assume towards a contradiction that 1729 is prime. Look at
If this were prime, we would have that the only two solutions to this equation are given by and (if you haven't seen this before, isolate for 0, factor and argue that because we've assumed that 1729 is prime, it has to divide one of the factors). However, notice that from the given congruences, we have that but
and this is a contradiction. Hence 1729 is not prime. 