Can every positive integer be expressed as a difference between integer powers?

Question

In mathematical notation, I am asking if the following statement holds:

$$\forall\,n>0,\,\,\exists\,a,b,x,y>1\,\,\,\,\text{ such that }\,\,\,\,n=a^x-b^y$$

A few examples:

• $1=9-8=3^2-2^3$
• $2=27-25=3^3-5^2$
• $3=128-125=2^7-5^3$

I wrote a small Python script to compute a handful of examples:

powers = []
for i in range(2,8):
    for j in range(2,8):
        power = i**j
        if power not in powers:
            powers.append(power)

diffs = []
for i in range(0,len(powers)):
    for j in range(i+1,len(powers)):
        diff = abs(powers[i]-powers[j])
        if diff not in diffs:
            diffs.append(diff)

print sorted(diffs)

The first few missing values are $6$, $10$ and $14$.

But it doesn't prove of course that no such $a,b,x,y$ exist for them.

How should I tackle this problem? Any links to related research will also be appreciated.

Answer 1

See OEIS sequence A074981 and references there. $10$ does have a solution as $13^3-3^7$, but apparently no solutions are known for $6$ and $14$.

Answer 2

This is very probably an open problem. It is closely related to Pillai's conjecture which is a generalization of Catalan's conjecture. The former looks for solutions in the integers to

$$ a^n - b^m = c$$

Catalan's conjecture is that $a=m=3$ and $b=n=2$ is the only solution for $c=1$. It was proven in April 2002 by Pedra Mihăilescu.

EDIT: @RobertIsrael found the conjectured list of values of $c$ that give no solution to Pillai's equation above.