Given N>0 and
N=PPPP.....PP(1998 digits)
what is the thousand digit after then decimal point of sqrt(N)????
Nobody else has attempted this, so I'll have a shot at it.
Start with the case where P = 9. Let N = 999...99 (2n digits). Eventually we'll want n = 999, but it's easier to start with a general n. Then , and so
. . . . .
So the (n+1)th digit after the decimal point is a 4. In particular, if n = 999 then the thousandth digit after the decimal point is a 4.
That deals with the case when P = 9. If P = 1, then you can divide by 3 to see that the 1000th digit of (1998 digits) is a 1. If P = 4 then the 1000th digit is a 3. But for other values of P, when is irrational, I doubt whether there is any straightforward analytical way to answer the question.