Fractional Parts of xⁿ

affine·mess

x = ≈ 1.6180339887498948482…

n to

Color dots by

Y axis shows

Examples

φ=(1+√5)/2 ρ (plastic) 1+√2 √2 √3 ∛2 3/2 4/3 32/17 e π (√2+√3)/2 2^(1/5)

Ops: + − * / ^ · sqrt(x) cbrt(x) root(n,x)
Constants: pi, e, phi. Enter computes.

Fractional parts of xⁿ

Pick a number x > 1 and look at the fractional parts of x, x², x³, … Each dot is one power: n across, the part after the decimal point up.

Almost always: noise

For almost every x > 1, the fractional parts of xⁿ are equidistributed in [0, 1): in the long run every interval gets its fair share (Koksma, 1935). Try √2, ∛2 or 32/17 and the dots fill the strip evenly.

Sometimes: not at all

Try φ, 1+√2 or the plastic ratio ρ. The dots collapse onto 0 and 1, because xⁿ gets closer and closer to an integer. These are Pisot numbers: algebraic integers whose other conjugates all have absolute value below 1, so their contribution to xⁿ dies away. Switch the y-axis to "Dist. to ℤ" to watch it go to zero, and colour by parity to see which side it approaches from.

And nobody knows

For 3/2, e or π, nobody can prove anything of the kind. It is not even known whether the fractional parts of (3/2)ⁿ are dense in [0, 1]. The picture looks perfectly random; the theorem is wide open.

Under the hood

Powers are computed in high precision with decimal.js, and the working precision grows with n so the fractional part is still right at n = 1000. Plot by Plotly.