all prime numbers have irrational square roots

Question

How can I prove that all prime numbers have irrational square roots?

My work so far: suppose that a prime p = a*a then p is divisible by a. Contradiction. Did I begin correctly? How to continue?

The work so far is not correct. You need to suppose that $p=\left(\frac{a}{b}\right)^2$, where $a$ and $b$ are integers (with naturally $b\ne 0$). Probably you will also want to say that without loss of generality $a$ and $b$ have no common factor greater than $1$. — André Nicolas, Dec 05 '13 at 18:04

score 7 · Accepted Answer · answered Dec 05 '13 at 18:00

7

The standard proof for $p=2$ works for any prime number.

answered Dec 05 '13 at 18:00

Igor Rivin

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How (I see that it does, but why formally does that work?) – Alec Teal Dec 05 '13 at 18:01
@AlecTeal Because $p\mid a^2\implies p\mid a $ for $p$ a prime. – Pedro Dec 05 '13 at 18:02
Sure, just so simply write up infinitely many proofs.... – Adam Dec 05 '13 at 18:04
@Adam What...? ${}{}{}$ – Pedro Dec 05 '13 at 18:09
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@PedroTamaroff I think Adam should write up infinitely many proofs -- he might learn something. – Igor Rivin Dec 05 '13 at 18:10
@PedroTamaroff: I (mistakenly) thought that Igor is suggesting to prove the result for every prime number separately.... – Adam Dec 05 '13 at 18:29
@Adam I am glad you figured things out. – Igor Rivin Dec 05 '13 at 18:30

score 5 · Answer 2 · answered Dec 05 '13 at 18:08

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Note that this answer is included only so it is clear what Igor Rivin meant by standard proof:

$$\sqrt p = \frac a b$$ where $a$ and $b$ are co-prime (the fraction is reduced) and $p$ is prime. $$b^2 p = a^2$$ as $a^2$ must disible by $p$ means $a$ must be divisible by $p$ $$b^2 = p c^2$$ where $c=\frac p a$

This will result in the fact that $b$ must also be divisble by $p$ which is a contraction of the fact that $a$ and $b$ are co-prime.

Please try this with $p = 4$ and see where is breaks down.

answered Dec 05 '13 at 18:08

kaine

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We assumed what? – Adam Dec 05 '13 at 18:13
@kaine I misread something. I apologize. =) – Pedro Dec 05 '13 at 18:15
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Adam, the assumptions are that p is prime and the $\frac a b$ is in it's simplist form, aka it cannot be $\frac 4 2$ as that would reduce to $\frac 2 1$. – kaine Dec 05 '13 at 18:16
I think Kaine's proof is correct. – Adam Dec 05 '13 at 18:20
@Adam you should probably accept Mr. Rivin answer then, my answer is just clarification of his in case it was not clear what he meant. – kaine Dec 05 '13 at 18:23
Alternately use RRT on $x^2-p$. – tc1729 Dec 05 '13 at 18:27
What does RRT mean? – Igor Rivin Dec 05 '13 at 18:35
http://en.wikipedia.org/wiki/Rapidly_exploring_random_tree – kaine Dec 05 '13 at 18:36
@kaine ah, that explains it :) – Igor Rivin Dec 05 '13 at 18:38
Hmm not sure if the above is sarcasm, but I meant Rational Root Theorem. – tc1729 Dec 05 '13 at 22:27

score 1 · Answer 3 · answered Dec 05 '13 at 19:00

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$p=a^2/b^2\implies a^2=pb^2$. The number of prime factors of $a^2$ is even whereas the numbers of prime factors of $ pb^2$ is odd.

answered Dec 05 '13 at 19:00

Michael Hoppe

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score 0 · Answer 4 · answered Dec 05 '13 at 18:04

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Your attempt only shows that the square root of a prime is not an integer.

answered Dec 05 '13 at 18:04

mrf

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I am aware of the fact. – Adam Dec 05 '13 at 18:04

all prime numbers have irrational square roots

4 Answers4