If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers

Question

Problem:

If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers

I've come to the conclusion that $a \ne b$ and $a \ne x$ and $b \ne x$ but I guess that won't really help me.. I understand that if the $\gcd$ between two numbers if $1$ then they obviously have no common divisors but where do I go from this point?

Any tips at tackling this would be great. It looks quite easy though I'm still trying to get my hand around these proofs! Any pointers in the right direction would be great.

Thank you in advance,

Answer 1

Below is an approach using basic gcd arithmetic (associative, commutative, distributive laws), some of which you may need to (simply) prove before you can use this method (once done, you will gain great power). Below we explicitly show that $\rm\:a,b\:$ are squares by taking gcds. Namely

Lemma $\rm\ \ \color{#0a0}{(a,b,c) = 1},\,\ \color{#c00}{c^2 = ab}\ \Rightarrow\ a = (a,c)^2,\,\ b = (b,c)^2\ $ for $\rm\:a,b,c\in \mathbb N$

Proof $\rm\ \ (a,c)^2 = (a^2,\color{#c00}{c^2},ac) = (a^2,\color{#c00}{ab},ac) = a\color{#0a0}{(a,b,c)} = a.\ $ Similarly for $\,\rm(b,c)^2.\ \ $ QED

OP is the special case $\rm\:(a,b) = 1\ (\Rightarrow\ (a,b,c) = 1)$.

Generally $\rm\: \color{#c00}{ab = cd}\: \Rightarrow\: (a,c)(a,d) = (aa,\color{#c00}{cd},ac,ad) = a\:\!(a,\color{#c00}b,c,d) = a\:$ if $\rm\:(a,b,c,d)\! =\! 1.\:$ For more on this and closely related topics such as Euler's four number theorem (Vierzahlensatz), Riesz interpolation, or Schreier refinement see this post and this post.

The Lemma easily generalizes from squares to $n$'th powers - see here.

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Compare the following Bezout-based proof (this is a simplified form of the proof in Rob's answer).

Note that $\ 1=\overbrace{a{\rm u}+b\,{\rm v}}^{\large (a,b)}\,\ \overset{\large \times\,a}\Rightarrow\ a = \color{#c00}{a^2}{\rm u}+\!\!\overbrace{ab}^{\Large\ \ \color{#c00}{c^2}}{\rm v} \ \,$ so $\,\ d=(a,c)\mid a,c\,\Rightarrow\, d^2\!\mid \color{#c00}{a^2,c^2}\,\Rightarrow\, d^2\!\mid a$

Conversely $\ d = (a,c)= au+cv\,\ \Rightarrow\,\ d^2=\,\color{#c00}{a^2}u^2+2\color{#c00}acuv+\color{#c00}{c^2}v^2\ \ $ thus $\ \ \color{#c00}{a\mid c^2}\ \Rightarrow\,\ \color{#c00}a\mid d^2$

$\quad\ \ {\rm i.e.}\quad (d)= (a,c)\ \ \Rightarrow\ \ (d^2) \:\!\subseteq\, (a,c^2)\,\color{#0a0}{\subseteq\, (a)}\ \ $ by $\ \ a\mid c^2\,\ $ [simpler ideal form of prior]

Simpler, $ $ via above Lemma: $\,\ (d)^2 =\, (a,c)^2 \color{#0a0}{=\, (a)}.\,$ Note how this ideal / gcd version eliminates the obfuscatory Bezout coefficients $\,u,v,\,$ and allows us to simultaneously prove both directions .

Answer 2

Fundamental theorem of arithmetic says that every number has a unique prime factorization.

If gcd(a,b) = 1, then all of these factors are unique (no prime factor is shared between a and b). What does this say about $x^2$?

Hint 2:

Let $a_i$ be a prime factor of a and $b_i$ be a factor of b. Then,

$$ab = \prod {a_{i}^{m_i}}\prod {b_{i}^{n_i}}$$

But $x$ has to have a unique factorization in the form,

$$ x = \prod {x_{i}^{e_i}} $$, where $m, n, e$ are integer exponents. Keep in mind it is unique and we can order these factors in any way we please. What does this say about $x^2$ compared to $ab$?

Answer 3

Here is a proof using Bezout's Identity.

Let $x^2=ab$ and $\gcd(a,b)=1$, where $a,b\gt0$.

There are $u,v$ so that $$ au+bv=1\tag{1} $$ Let $s_a=\gcd(x,a)$. Rewriting $(1)$, we have $$ \begin{align} s_a\left(\dfrac{a}{s_a}\right)u+bv=1 &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2=1-b(2v-bv^2)\tag{2}\\ &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2a+ab(2v-bv^2)=a\tag{3}\\ &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2a+s_a^2\left(\dfrac{x}{s_a}\right)^2(2v-bv^2)=a\tag{4}\\[8pt] &\implies s_a^2\mid a\tag{5} \end{align} $$ Justification:
$(2)$: Move $bv$ to the right side and square
$(3)$: Move $b(2v-bv^2)$ to the left side and multiply by $a$
$(4)$: $x^2=ab$
$(5)$: $s_a^2$ divides each term on the left side

There are $u_a,v_a$ so that $$ \begin{align} xu_a+av_a=s_a &\implies\frac{x}{s_a}u_a+\frac{a}{s_a}v_a=1\tag{6}\\ &\implies\frac{x^2}{s_a^2}u_a^2=1-\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)\tag{7}\\ &\implies\frac{ab}{s_a^2}u_a^2+\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)=1\tag{8}\\ &\implies abu_a^2+as_a\left(2v_a-\frac{a}{s_a}v_a^2\right)=s_a^2\tag{9}\\[7pt] &\implies a\mid s_a^2\tag{10} \end{align} $$ Justification:
$\ \:(6)$: Divide by $s_a$
$\ \:(7)$: Move $\frac{a}{s_a}v_a$ to the right side and square
$\ \:(8)$: Move $\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)$ to the left side, $x^2=ab$
$\ \:(9)$: Multiply by $s_a^2$
$(10)$: $a$ divides each term on the left side

Combining $(5)$ and $(10)$ yields $a=\gcd(x,a)^2$. Symmetry yields, $b=\gcd(x,b)^2$.