I am studying my conjecture that $2^n$ and $2^{n+1}$ have a common digit in base 5 if $n>6$. I believe that this conjecture is true provided that

$$ 2^x=12(5^{y_1}+ \dots +5^{y_k}) + 5^z - 1 $$

has no solutions in integers where $x \ge 0$, $0\le y_1 < \dots <y_k$, and $z \ge 0$. This arises from the fact that, in base 5, $2 \times 1 = 2$ and $2 \times 14_5 = 33_5$; so that for example $2 \times 114111141411_5 = 233222333322_5$. The first difficult base is 5: bases 2 and 4 are trivial and base 3 can be solved.

This is similar to the unsolved Erdos ternary conjecture that involves

$$ 2^x=3^{y_1}+ \dots +3^{y_k}. $$

My question is whether these two exponential Diophantine equations are related so is equally difficult to solve, or whether other techniques (such as modular considerations) can help to solve it.



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