1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I How to prove that 2n=1 has no integer solutions

  1. Feb 19, 2017 #1
    This might seem like a very simple problem, because we could just say that the only possible solution is n = 1/2, which is not an integer. But I am curious as to how to prove that there is no solution, with no knowledge of rational numbers, just as we can prove that x^2 = 2 as no rational solutions without any knowledge of irrational numbers.
     
  2. jcsd
  3. Feb 19, 2017 #2
    I am confused. 1/2 is never an integer or even number. I am asking how I would prove that 2n=1 has no integer solutions, without just saying that n=1/2 is not an integer.
     
  4. Feb 19, 2017 #3

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    There are different approaches. For example, derive (from a chosen set of integer axioms) that n solving this equation must be both even and odd, a contradiction.
     
  5. Feb 19, 2017 #4

    fresh_42

    Staff: Mentor

    The first thing to do - as always - check the statement: What is an integer? The proof of the irrationality of ##\sqrt{2}## which you mentioned, uses, that rationals can be written as quotients of integers, which are products of primes.

    If you take the definition of primes again, then ##2n=1## implies the prime ##2\,\vert \,1## and is therefore a unit. But primes aren't allowed to be units, so ##2 \nmid 1## and ##n \notin \mathbb{Z}\,.## Or if you like: the only units in ##\mathbb{Z}## are ##\pm 1## so it's impossible because of that.

    I know this is a bit like cheating, because it plays with definitions, so again: what is an integer?
     
  6. Feb 19, 2017 #5

    Mark44

    Staff: Mentor

    I deleted my earlier post, and PAllen's reply to it.
     
  7. Feb 19, 2017 #6

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    Or even simpler, along this line, it implies 1 is even; but 1 is odd.
     
  8. Feb 20, 2017 #7

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    With inequalities: 2*0 is not 1 (skip this if you don't include 0 in the integers). For every integer n larger than 0, ##2n \geq 2 > 1##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: How to prove that 2n=1 has no integer solutions
  1. Simplify (2n-1)! (Replies: 11)

Loading...