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# Is the number a square?

Find all pairs (m,n) of positive integers for which the above expression is a perfect square.

Note by Shashank Rammoorthy
2 years, 10 months ago

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Suppose $$2^{m}+3^{n}=a^{2}$$.

It is easy to show that both $$m$$ and $$n$$ are even.Let $$m=2r$$ and $$n=2s$$

Hence, $$2^{2r}=a^{2}-3^{2s}=(a-3^{s})(a+3^{s})$$

Hence, $$a-3^{s}=2^{i}$$...$$(1)$$ and $$a+3^{s}=2^{2r-i}$$...$$(2)$$

$$(2)-(1)$$ gives $$2.3^{s}=2^{i}(2^{2r-2i}-1)$$, which implies $$i=1$$.

Thus, $$a-3^{s}=2$$ and $$a+3^{s}=2^{2r-1}$$.Hence, $$3^{s}=2^{2r-2}-1$$...$$(3)$$

Suppose, $$s>1$$.Then $$r≥3$$.But then the equation $$(3)$$ is impossible since when divided by $$8$$, the left hand side $$3^{s}$$ leaves a remainder $$1$$ or $$3$$ while the right hand side would leave the remainder $$7$$.Thus $$s=1$$ is the only possibility.When $$s=1$$,i.e, $$n=2$$,we have the solution $$2^{4}+3^{2}=25$$.Thus $$(m,n)=(4,2)$$ is the only solution.

- 2 years, 10 months ago

How to show that both $$m$$ and $$n$$ are even?

- 2 years, 10 months ago

$$a^{2}≡0 or 1(mod 3)$$ and $$2^{m}+3^{n}≡2^{m}(mod 3)$$.But $$2^{m}$$ is not congruent $$0$$ modulo $$3$$.

So,$$2^{m}≡1(mod 3)$$ which implies $$m$$ is even.Hence, $$3^{n}≡a^{2}≡0 or 1 (mod 4)$$.But $$4$$ does not divides $$3^{n}$$.So,$$3^{n}≡1(mod4)$$ which implies $$n$$ is even.

- 2 years, 10 months ago

Thank you for this solution! Been thinking about this all day. Never got the chance to sit down with a pen and paper, unfortunately...

- 2 years, 10 months ago

Could you please explain your solution from line 5 onwards? How is it 2.3^s? Thanks.

- 2 years, 10 months ago

I just subtracted eqn $$1$$ from eqn $$2$$ and got $$2\times3^{s}=2^{2r-i}-2^{i}=2^{i}(2^{2r-2i}-1)$$.

Hence, $$3^{s}=2^{i-1}(2^{2r-2i}-1)$$.If $$i>1$$,then $$2$$ divides $$3^{s}$$, which is impossible.So $$i=1$$.

So,$$3^{s}=2^{1-1}(2^{2r-2.1}-1)=2^{2r-2}-1$$, which is eqn $$3$$.Then the solution is very clear.

- 2 years, 10 months ago

Thanks.

- 2 years, 10 months ago

How about $$(3,0)$$ and $$(0,1)$$ ?

- 2 years, 10 months ago

m, n are positive integers. 0 is not positive.

- 2 years, 10 months ago

(4,2),(3,0),(0,1),

- 2 years, 10 months ago