# [EM] 120 Seats

Bob Richard [electorama] electorama at RobertJRichard.com
Mon Apr 17 08:29:10 PDT 2006

```In "Seats and Votes" (1989), Chapter 15, Taagepera & Shugart document
the empirical generalization that the size of the lower house tends to
vary with the cube root of the population.  They also develop a
theoretical model to explain why this might be true.  Essentially, the
cube root of two times the number of voters is size that minimizes the
sum of time spent on constituent communication plus time spent on
colleague communication for the individual legislator.  Strictly
speaking, the theory only applies to single member districts but the
empirical generalization appears to be broader than that.

How many registered voters are there in Israel?

--Bob Richard

-----Original Message-----
From: election-methods-bounces at electorama.com
[mailto:election-methods-bounces at electorama.com]On Behalf Of Doreen
Dotan
Sent: Sunday, April 16, 2006 2:41 PM
To: election-methods at electorama.com
Subject: [EM] 120 Seats

I am asking the following question in the framework of an attempt to
write a proposal for a revision of the electoral system in Israel.

There seems to be a large number of parliaments and legislatures that
are composed of 120 seats in countries that are very dissimilar
demographically, politically, economically, geographically and so on.

Is the number 120 significant because of the particular mathematical
properties of the number 120, to wit:

120 is the factorial of 5. It is the sum of a twin prime pair (59 + 61)
as well as the sum of four consecutive primes (23 + 29 + 31 + 37). It is
highly composite, superabundant, and colossally abundant number, with
its 16 divisors being more than any number lower than it has, and it is
also the smallest number to have exactly that many divisors. It is also
a sparsely totient number. 120 is the smallest number to appear six
times in Pascal's triangle, and it is also a Harshad number.
It is the eighth hexagonal number and the fifteenth triangular number,
as well as the sum of the first eight triangular numbers, making it also
a tetrahedral number.
120 is the first multiply perfect number of order three (a 3-perfect
number). The sum of its factors (including one and itself) sum to 360;
exactly three times 120. Note that perfect numbers are order two
(2-perfect) by the same definition.
120 is divisible by the number of primes below it, 30 in this case.
However there is no integer which has 120 as the sum of its proper
divisors, making 120 an untouchable number.
120 figures in Fermat's modified Diophantine problem as the largest
known integer of the sequence 1, 3, 8, 120. Fermat wanted to find
another positive integer that multiplied with any of the other numbers
in the sequence yields a number that is one less than a square. Euler
also searched for this number, but failed to find it, but did find a
fractional number that meets the other conditions, 777480 / 287922.
The internal angles of a regular hexagon (one where all sides and all
angles are equal) are all 120 degrees.

Source: http://en.wikipedia.org/wiki/120_(number)

or is there some other reason? Are there additional reasons?

Doreen Ellen Bell-Dotan, Tzfat, Israel

http://www.geocities.com/dordot2001/ImperativeOfMoralMaths.html

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