I am searching form perfect squares of the form : $$1\underbrace{2\dots2}_a\underbrace{3\dots3}_b\underbrace{4\dots4}_c\underbrace{5\dots5}_d\underbrace{6\dots6}_e\underbrace{7\dots7}_f\underbrace{8\dots8}_g9,$$ where $(a,b,c,d,e,f)\in (\mathbb{N^*})^6$, and $g\in\mathbb{N}$.
With a systematic search using python and gmpy I could not find a single number with $a+b+c+d+e+f+g<50$ digits. Is there an argument that prevent such perfect squares from existing?
What if we allow numbers of the form : 12...23..34..45..56..67..78..8912...23..34..45..56..67..78..89?