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If p is prime, then its square root is irrational

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Homework Statement



Im trying to prove that if p is prime, then its square root is irrational.


The Attempt at a Solution



Is a proof by contradiction a good way to do this?

All i can think of is suppose p is prime and √p is a/b,

p= (a^2)/ (b^2)
Is there any property i can exploit to go on or is should i attempt other methods of proof
(contrapositive, direct, induction?)
 

Answers and Replies

  • #2
Dick
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Homework Statement



Im trying to prove that if p is prime, then its square root is irrational.


The Attempt at a Solution



Is a proof by contradiction a good way to do this?

All i can think of is suppose p is prime and √p is a/b,

p= (a^2)/ (b^2)
Is there any property i can exploit to go on or is should i attempt other methods of proof
(contrapositive, direct, induction?)
You can assume a and b have no common factors, right? Go for a contradiction. a must be divisible by p. Can you show that?
 
  • #3
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attempt

I think we need to prove that no prime is square.

This makes sense in my head, but I can't seem to figure it out!

By the way, is there a theorem that says that square roots of non square numbers are irrational?
 
  • #4
HallsofIvy
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Yes, "no prime is a square" is exactly what "if p is a prime then it is not a square" says. If you "can't seem to figure it out", then look at the specifice words of the definitions of "prime" and "square".

Then do an indirect proof as Dick suggested. Suppose there exist a prime, p, that is a "square". Then [tex]p= n^2[/tex] for some integer n.
 
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  • #5
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I think its obvious that a prime number cant be a square of an integer (trivial by definition), but that does not imply it cannot be a square of a rational. The thing im trying to prove is that the square root of primes ,is not rational. Am i misunderstanding the logic ,overlooking or ignorant of something that i can use to advance in the proving?
 
  • #6
D H
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What does the rational root theorem have to say about x2-p=0, where p is a prime number?
 
  • #7
Dick
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I think its obvious that a prime number cant be a square of an integer (trivial by definition), but that does not imply it cannot be a square of a rational. The thing im trying to prove is that the square root of primes ,is not rational. Am i misunderstanding the logic ,overlooking or ignorant of something that i can use to advance in the proving?
You didn't pay much attention to the hint I gave in post 2. So I won't repeat it.
 
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  • #8
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You didn't pay much attention to the hint I gave in post 2. So I won't repeat it.
Well i dont really know how to use the hint. But anyway its been explained in my online course how to prove it using the fact that sqrt of 2 and 3 are irrational, and using it to generalise it to primes( we did the proofs for sqrt of 2 and 3 earlier in the course). Thanks for the help guys ,its very much appreciated.
 

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