Every Fifth Real Number is Evil

Every Fifth Real Number is Evil
By Doron Zeilberger
.pdf .tex
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org .

Written: Oct. 4, 2025

In fond memory of my beloved cousin Matti Weiss (1945-2025) who ignited my love of mathematics

Fifteen years ago, then-Carleton-undergrad Isaac Hodes, proved that the Golden Ratio is evil. In this modest contribution to human knowledge, we show that in fact, every fifth real number is evil, and we present lots of other interesting numbers that are evil. We also show (in addition to many other fascinating facts), that the expected evil-location of a random evil number is 148.185185185185185... . It follows that the Golden Ratio is a fairly average evil real number, since, as first computed by Hodes, the evil location of the Golden Ratio is 146.
This note is dedicated to the memory of my first-cousin, Matti Weiss (1945-2025), who (when I was 12 and he was 16) infected me with the love of mathematics.

Added Oct. 10, 2025: Isaac Hodes told me about this interesting page, by Ed Pegg, Jr. and Chris Lomont. so Theorem 1 was already known to them, written in 2004. So I was scooped by them regarding the "one in five real numbers is evil" fact. However Theorems 2 and Theorem3 in my paper are hopefully new.

Added Oct. 27, 2025: Read interesting comments by Jordan Stoyanov.

Maple package

Hodes.txt, a Maple package to investigate evil numbers (and their frequency)

Statistical Analysis

If you want to see rigorously proved, closed-form expression for the expectation, variance, and the first 16 central moments of the random variable "evil-location" for a random base-b, evil real number (for an arbitrary, i.e. symboilc b)
The input file gives the output file.

Sample Input and Output Files for Hodes.txt: Specific families

If you want to see all the prime numbers p, amongst the first 10000 prime numbers, such that p*π is evil, along with the corresponding evil locations
The input file gives the output file.
As you can see at the bottom, their frequency is .20105 (not far from those for all real numbers, that is 0.2)

If you want to see all the (irrational) BASE 10 evil square-roots-of integer under 10000
The input file gives the output file.
As you can see at the bottom, their frequency .1966666667 (not far from those for all real numbers, that is 0.2)

If you want to see all the (irrational) BASE 3 evil square-roots-of integer under 10000
The input file gives the output file.
As you can see at the bottom, their frequency 0.66 (not far from those for all real numbers, that is 2/3)

If you want to see all the (irrational) BASE 13 evil square-roots-of integer under 10000
The input file gives the output file.
As you can see at the bottom, their frequency is 0.1536363636 (not far from those for all real numbers, that is 2/13=0.1538461538)

If you want to see all the BASE-10 evil numbers of the form Pi*sqrt(x), for x between 1 and 10000,
The input file gives the output file.
As you can see at the bottom, their frequency is .2045 (not far from those for all real numbers, that is 0.2)

If you want to see all the BASE-3 evil numbers of the form Pi*sqrt(x), for x between 1 and 10000,
The input file gives the output file.
As you can see at the bottom, their frequency is 0.6691 (not far from those for all real numbers, that is 2/3)

If you want to see all the BASE-13 evil numbers of the form Pi*sqrt(x), for x between 1 and 10000,
The input file gives the output file.
As you can see at the bottom, their frequency is 0.1546000000 (not far from those for all real numbers, that is 2/13= 0.1538461538)

If you want to see all the BASE-10 evil cubic-roots of x (for non-perfect-cubes x between 1 and 10000),
The input file gives the output file.
As you can see at the bottom, their frequency is .1962120453 (not far from those for all real numbers, that is 0.2)

If you want to see all the BASE-3 evil cubic-roots of x (for non-perfect-cubes x between 1 and 10000),
The input file gives the output file.
As you can see at the bottom, their frequency is 0.6657981762 (not far from those for all real numbers, that is 2/3)

If you want to see all the BASE-13 evil cubic-roots of x (for non-perfect-cubes x between 1 and 10000),
The input file gives the output file.
As you can see at the bottom, their frequency is .1540234492 (not far from those for all real numbers, that is 2/13= 0.1538461538...)

If you want to see all the BASE-10 evil numbers of the form cos(x)² for x between 1 and 10000
The input file gives the output file.
> As you can see at the bottom, their frequency is 0.1973 (not far from those for all real numbers, that is 0.2)

If you want to see all the BASE-3 evil numbers of the form cos(x)² for x between 1 and 10000
The input file gives the output file.
As you can see at the bottom, their frequency is 0.6793 (not far from those for all real numbers, that is 2/3)

If you want to see all the BASE-13 evil numbers of the form cos(x)² for x between 1 and 10000
The input file gives the output file.
As you can see at the bottom, their frequency is 0.1544 (not far from those for all real numbers, that is 2/13=0.1538..)

Personal Journal of Shalosh B. Ekhad and Zeilberger