Juggler Sequences

In number theory, a juggler sequence is an integer sequence that starts with a positive integer x₀ and that evolves recursively as follows:
x₁ = floor(x₀^0.5) if x₀ is even
x₁ = floor(x₀^1.5) if x₀ is odd

The conjecture, put forward by C.A. Pickover, is that all sequences eventually reach 1 (see Wikipedia). The conjecture has not been proven thus far. It has been numerically verified for x₀ values of up to 7110200, using a tailored Python script. In this range, x_i values reach up to 10^8'000'000. Astonishingly high numbers... The first value, for which convergence has not been shown yet, is 7110201.

Mathematicians that were able to demonstrate convergence of 7110201 are welcome to submit their finding!

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Last update: 2026-05-12