I'm looking at a problem set question that asks me to find all the $6$-digit numbers ABCDEF that satisfy the condition that $2ABC=DEF$ (so for example, $100200$). They also need to satisfy the condition that $ABCDEF+1$ is a perfect square.
I've tried working on the problems from different ways, but I've come up with nothing.
My most recent attempt was to break $ABCDEF$ up so that $10^5A+10^4B+\dots+F$ became my new number. Then, I substituted so that I got
I was hoping that I could do something with digital roots to get the values of perfect squares such that $dr[1002(100A+10B+C)+1] = 1$ or $4$ or $7$ or $9$ ($1,4,7$ and $9$ being the digital roots that yield perfect squares), but it doesn't look easy and I'm pretty lost. I believe the question has to do with Number Theory, but I'm not sure since the class I'm in is fairly disorienting.
Any help would be appreciated.