Beautiful number theory numeral system problem.

Question

The product of two consecutive positive integers was taken and converted into some numeral base-n system.

In this base-n system it's written as a double-digit number with consecutive positive integer digits, each at most 9. Find these digits.

I tried to find these numbers and their digits by checking. Do we have to check all satisfying bases(base-1,base-2,...,base-9) and combine? How is it done?

Answer 1

The number we are looking for is

1. of the form $x(x+1)\ = x^2+x$ by the first criterium, and
2. of the form $y\cdot n+(y+1)\ = (n+1)y+1$ by the second criterium, with $y<9$ and $y+1<n$.

So, basically take an $x$ and consider a factorization of $A:=x^2+x-1$ that satisfies 2.

E.g. for $x=1$ we get $A=1=1\cdot 1$ which gives $y=1$ and $n=0$ (which is not valid, $n$ should be $\ge2$).
For $x=2$, we get $A=5=1\cdot 5$ whence you can take $n=4$ and $y=1$ (and, indeed, $2\cdot 3=6=[12]_{(4)}$).
And so on...