The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \frac{10^a+10^{2a}+1}{3} \cdot \frac{10^b+10^{2b}+1}{3}$$ is palindromic in base $10$. This conjecture was experimented well.
Example: taking $a=2,b=3,c=5$, we will get $$3367 \cdot 333667 \cdot 99999=112344555443211$$ which is palindromic in base $10$, my question is to prove or disprove this conjecture. Note that the conjecture above was proved before editing . another conjecture that I ask to prove or disprove it is that the sequence of the numbers of the form (10^a+10^2a+1)/3 is the maximally dense sequence in base 10 with the palindromic products property as described above .
