MATH 200304202159

Index: neobabylonic nrs ,
complex nrs ,
recursive nrs
## Little Mathematical Items

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### The Neobabylonic Numbers

- ( Leo Horowitz , 1987/10/21 )
- The Babylonians, they chose as the base of their calculations
numbers like 12,60,360 . It appears that these belong to seven numbers
with a special property concerning divisibility .

- Consider the natural numbers 1,2,..

- Definition. An "antiprime number" is a number that has
more dividers than all its predecessors .

- Definition. A "neobabylonic number" is an (antiprime)number
such that the next bigger antiprime number is (at least) twice as big .

- Hypo-thesis. There are exactly seven neobabylonic numbers, being: 1 2 6 12 60 360 2520 .

- (2023/09/22) I never found a bigger neobabylonic number.

Anyway, here is a method to calculate the number of dividers of a given number :

First write the given number as a product of (highest possible) powers of primenumbers. Then add one to the power-numbers and multiplicate them:

Nr. of dividers of a number p1^q1*...*pn^qn (p=primenr., ^q=tothe power), is (q1+1)*...*(qn+1) .

E.g. 360= 5*3*3*2*2*2, so 2*3*4 dividers = 24 dividers.

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### Theory of Complex Numbers

A proof of the "**Fundamental Theorem of Algebra**"
by means of galois theory and 2-sylowgroups .

- (L. Horowitz , 1965 ; conceived in 1959 after following
a course of prof. H.D. Kloosterman at Leiden )
- Theorem. If R is the field of real numbers , and if i*i=-1 ,
then R(i) is algebraically closed .

- The little article has appeared in Nieuw Archief voor Wiskunde (3) ,
XIV , page 95-96 (1966)

- The same proof has appeared independently as an exercise
in a book of E. Artin "Algebra" in 1965 .

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### Theory of Enumerable Sets

Invariance of partial order of **Recursive Equivalence Types**
under finite division .

- ( L. Horowitz, 1965/09/10 communicated to
prof A. Nerode )
- Theorem. If n > 0 and A and B are R.E.T.'s
then nA __>__ nB is equivalent to A __>__ B .

- It it proved here only using conventional proof-methods ,
and obtained during a seminar of prof B. van Rootselaar at Amsterdam .

- It appeared in Indagationes Mathematicae Amsterdam , 29 , no 1
(1967) 8 pp

- It has also been proved by A. Nerode using the
priority method , to appear in Mathematische Annalen 1966 .

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