Help to solve exponential equation over positive integers

Question

I am trying to solve the equation

$\frac{4}{9} = \big( \frac{1}{3} \big) ^n + \big( \frac{1}{3} \big) ^m $

with $n$ and $m$ both positive integers.

I type into Mathematica the following

In:= Solve[4/9 == 1/3^m + 1/3^n && m > 0 && n > 0, n, Integers]

which spits out "Solve::nsmet: This system cannot be solved with the methods available to Solve."

The solution to the problem is $n=1\:\&\:m=2$ or $n=2\:\&\:m=1$. How can I change my input to have Mathematica come up with this?

m_goldberg · Accepted Answer · 2014-01-25 03:26:09Z

7

One approach is to ask for an answer using FindInstance

FindInstance[4/9 == (1/3)^m + (1/3)^n && m > 0 && n > 0, {n, m}]

{{n -> 2, m -> 1}}

As Nasser points out in the comment, you can replace the requirements that m>0 and n>0 by insisting that n and m are integers:

FindInstance[(4/9) == (1/3)^n + (1/3)^m, {n, m}, Integers]

Both give the same answer.

m_goldberg

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answered Jan 24 '14 at 16:31

bill s

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$\begingroup$ fyi, FindInstance[(4/9) == (1/3)^n + (1/3)^m, {n, m}, Integers] also. $\endgroup$ – Nasser Jan 24 '14 at 16:43
$\begingroup$ Thanks! Had no idea that function existed. Mathematica is huge. $\endgroup$ – Brian Jan 24 '14 at 17:24
$\begingroup$ Solve, FindInstance, and Reduce are all devoted to this kind of thing. I always try out all three -- in this case, FindInstance was the most useful. $\endgroup$ – bill s Jan 24 '14 at 17:29

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