Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Math question



Someone on this list asked a question about a theorem that was proven
by Paul Erdos and Mark Kac. The information was in the latest issue of
Science. The theorem is: "...a graph of the number of prime factors of
very large numbers forms a bell curve ..." The statement is incomplete
because it does not say what is plotted along the abscissa. I can only
give an educated guess. I think it is a plot of the number of times a
given prime number appears in the prime-factor decomposition of a natural
number versus the prime number itself. For example, the number 120 has
the prime 2 appears three times, the prime 3 appears once and the prime
5 appears once. So, if my guess is right, very large numbers will have
a prime factor that will appear more than any other and then both smaller
prime factors and larger prime factors will occur with less frequency,
giving the bell-shaped curve.

I think this is likely incorrect. The plain language interpretation
would be that the number of very large numbers with N prime factors
is distributed normally over N. There is some average number Nav of
prime factors of very large numbers (say numbers distributed over a
defined interval, from 10**100 to 10**200) and there are more of
these with a number of prime factors near Nav than there are with
a number of prime factors either much greater than or much less
than Nav.

Just as we might say that a graph of heights of 60-year- old men
forms a bell curve, this locution seems to indicate that the
quantity mentioned (the number of prime factors) lies on the
independent variable, or x-axis. (I never could get "ordinate" and
"abscissa" straight.)

If that is the correct interpretation this result is interesting
but unsurprising.

Leigh