# What is a Juggler Number?

### ``` Juggler Sequence defined in Dr. Clifford A. Pickover's book "Computers and the Imagination." This book has scores of educational and recreational experiments that can be done on a personal computer. At the end of Appendix C in this book you will find: ". . . An award of 50 dollars is offered by the publisher for a printout of the largest Juggler number computed by readers. . . . Currently the largest juggler number computed s a 45,391-digit giant from the starting number 30817. It was computed by Harry J. Smith using hid own software package to perform multiple precision integer arithmetic. His package is written in the object-oriented programming language Turbo Pascal 5.5 by Borland International, Inc. His juggler package is a subset of his super-precision calculator software which computes transcendental functions to thousand of decimal places. Write him to obtain the software: . . ." Juggler Sequence also defined in Nov 1990 issue of Algorithm in PERSONAL PROGRAMS by Clifford A. Pickover. Pickover's definition of a juggler sequence: input positive integer, x repeat if x is even then x <-- [x^(1/2)] else x <-- [x^(3/2)] until x=1 The bracket signs indicate that non-integer numbers are truncated (i.e., 4.9 -> 4). A Juggler number is any number in a Juggler sequence. The Juggler Sequence starting with x(0) = 48443 has a max juggler number at x(60), a 972,463-digit number. Summary output of program JugglerC ----------------------------------- x(0) max with 1 digits for x(0) = 1 x(0) max with 1 digits for x(0) = 2 x(3) max with 2 digits for x(0) = 3 x(2) max with 3 digits for x(0) = 9 x(3) max with 5 digits for x(0) = 25 x(8) max with 14 digits for x(0) = 37 x(9) max with 27 digits for x(0) = 113 x(17) max with 82 digits for x(0) = 173 x(47) max with 271 digits for x(0) = 193 x(32) max with 5929 digits for x(0) = 2183 x(54) max with 8201 digits for x(0) = 11229 x(25) max with 11723 digits for x(0) = 15065 x(43) max with 23889 digits for x(0) = 15845 x(39) max with 45391 digits for x(0) = 30817 x(60) max with 972463 digits for x(0) = 48443 x(148) max with 1909410 digits for x(0) = 2,75485 x(99) max with 1952329 digits for x(0) = 12,67909 x(89) max with 2855584 digits for x(0) = 22,64915 x(67) max with 7996276 digits for x(0) = 58,12827 x(90) max with 41564193 digits for x(0) = 71,10201 not final x(115) max with 53322381 digits for x(0) = 71,10201 not final, this gave U Up started Numbers too large for FHT multiply, digits = 71096512 > 2^26 = 67,108,864 PARI gave x(119) max with 89981517 digits for x(0) = 7110201 not final, this gave *** if: length (lg) overflow ```

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