Are $144, 233, 377,$ and $17711$ the only Fibonacci Numbers that contains only two different digits(in base ten)?
I noticed that $144, 233, 377,$ and $17711$ are all Fibonacci Numbers and they contain only two different digits.
But are there any Fibonacci Numbers not listed above that contains only two different digits?
My instinct is that if a Fibonacci Number has only two different digits(for example, $2$ and $8$), then the sum of the last two Fibonacci Numbers must be “favored” to the desired number.
For example, if we want to obtain a Fibonacci Number that contains only $3$ and $5$, then one of the number must end with $3$ or $8$, because the last digit of $8+5$ is $3$, or either way, $9+8$ if there is “carry-on”.
When I say “contains only two different digits”, then it is numbers like $27722, 588588, 3939939, \cdots$.
I tried to check all Fibonacci Numbers up to $10^{1000}$ using Paridroid, but none of the Fibonacci Numbers that are not listed above contains only two different digits.
Note:
- I’am only considering all Fibonacci Numbers that have at least $3$ digits.