solution
Searching and checking only numbers ni with BAT((sqrt(ni),10000) = 9100 from the beginning I had found immediately 53 and 477, but my "quadrupleprecision floating point format" codes were not exact enough to compute BAT((sqrt(ni),N1) or BAT((sqrt(ni),N2).
So first of all I had to write routines for the octupleprecision floating point format (4 registers for one floating point number) (236 bits for the mantissa). Then I was able to see that
BAT((sqrt(53),N1) = BAT((sqrt(477),N1) and so on.
