Prove that you can formulate any whole integer using any function contaning the number four only three times.?

Question

Using four only three times in any function formulate any whole Integer: Example
1=4^(4 - 4)
2=(4+4) / 4

Answer 1

Of course. For any integer n:

n=-ln(ln (√√√√... √4)/ln 4)/ln 4

where the number of nested square root symbols is 2n

Why this works: √√4 = 4^(1/4), √√√√4 = 4^(1/4²), √√√√√√4 = 4^(1/4³), and in general √√√√√... √4 (2n times) = 4^(1/4^n). Therefore, -ln(ln (√√√√... √4)/ln 4)/ln 4 = -ln (ln(4^(1/4^n))/ln 4)/ln 4 = -ln (log_4 (4^(1/4^n)))/ln 4 = -ln (1/4^n)/ln 4 = -log_4 (4^(-n)) = -(-n) = n