By definition (10101)base b = b^4 + b^2 +1

and (10201)base b = b^4 + 2b^2 +1

Now b^4 + 2b^2 +1 = (b^2 + 1)^2 so it's composite for all real b.

But b^4 + b^2 +1 is a quadratic in b^2 which can't be factored in real numbers.

However, an Excel test up to b = 562 yields only composite integers for b^4 + b^2 +1. But how to prove it?