For some reason Plouffe seems to be ignoring me. Anyhow, here are some specific examples in various radices. (Incidentally, radices other than 2 such as 3, might make nicer image files.) in MAPLE9: f := (x) -> (4/Pi)*EllipticE(2*sqrt(x)) + ((8*x-2)/Pi)*EllipticK(2*sqrt(x)); which is a transcendental function. Why be algebraic when you can be transcendent? [I think it is known that f(rational) is going to yield transcendental numbers if not rational. I have not carefully verified that... theorems on this topic reviewed in this Waldschmidt paper: http://www.math.jussieu.fr/~miw/articles/pdf/SurveyTrdceEllipt2006.pdf ] evalf( f(10^(-10)), 60 ); 1.00000000010000000000250000000025000000003906250000765625000 evalf( f(10^(-100)), 6000 ); #first 6000 digits in radix 10 nonrandom evalf( f(10^(-1000)), 600000 ); #first 600000 digits in radix 10 nonrandom evalf( f(10^(-10000)), 60000000 ); #first 60000000 digits in radix 10 nonrandom etc. Note this verifies my earlier-posted claim the length of the nonrandomness goes as the SQUARE of N for f(10^(-N)). Of course if you prefer binary to base 10 then use f(2^(-100)) and the like. Recently on mathfun somebody mentioned the fact that the 5th-degree algebraic function y=f(x) defined as the solution y of the Bring-Jerrard irreducible quintic y^5 + y + x = 0 has a nice known Maclaurin series with integer coefficients which grow roughly simply-exponentially. Also that's mentioned here: http://en.wikipedia.org/wiki/Bring_radical#Series_representation So anyhow, why be a wimp and use 4th-degree or 2nd-degree algebraic numbers, when you can be cooler and use 5th-degree? So, MAPLE9 again... f := (x) -> solve( y^5 + y + x, y ); #warning, finds 5 roots, we only want the real root evalf( f(10^(-10)), 1400 ); #verifies the first 1400 digits are nonrandom evalf( f(10^(-100)), 140000 ); #longer nonrandom numbers evalf( f(10^(-1000)), 14000000 ); And basically the same thing works for ANY-degree algebraic numbers. For example, say you like Plouffe want degree=4? Try f := (x) -> solve( y^4 - y + x, y ); #warning, finds 4 roots, we only want the least real root evalf( f(10^(-10)), 1000 ); #verifies the first 1000 digits are nonrandom evalf( f(10^(-100)), 100000 ); #longer nonrandom numbers evalf( f(10^(-1000)), 10000000 ); Want degree=3? Try f := (x) -> solve( y^3 + y + x, y ); #warning, finds 3 roots, we only want the real root evalf( f(10^(-10)), 500 ); #verifies the first 500 digits are nonrandom Want quadratic? f := (x) -> sqrt(1-4*x); evalf( f(10^(-100)), 16000 ); evalf( f(10^(-1000)), 1600000 ); etc.