The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Inactive » math-history-list

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Take Another Look at = exp[ln(p1) + ln(p2) + ... + ln(pn)]
Replies: 7   Last Post: Mar 11, 2009 10:54 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Osher Doctorow Ph.D.

Posts: 30
From: Southern California
Registered: 3/10/07
Take Another Look at = exp[ln(p1) + ln(p2) + ... + ln(pn)]
Posted: Mar 9, 2009 11:59 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

>From Osher Doctorow

The Fundamental Theorem of Arithmetic tells us that every positive integer can be uniquely written as a product of powers of positive prime numbers up to the order of the factors, and to simplify notation let us assume that for a particular positive integer I we have:

1) I =, where 1, 2, ..., n are subscripts and pi are the primes in increasing or nondecreasing order with possible repetitions allowed.

This is equivalent to:

2) I = exp[ln(p1) + ln(p2) + ... + ln(pn)]

since of course exp(ln(x)) = x for x > 0.

Let us ask the question: does (1) which is overtly multiplicative or (2) which is overtly additive better characterize the arbitrary positive integer I?

Our first temptation is to say that since they are equivalent, I is both additive and multiplicative "fundamentally".

Let us then ask a second question: is the natural logarithm or the exponential function more "fundamental" for a positive integer I?

Again one is tempted to reply that they are equivalent in a sense in importance in (2) and so are intuitively at least equivalent in general.

There is, however, a curious thing about (2), namely that the logarithms not only occur additively, but that they are "internally non-decomposable" or "non-factorizable" since they are logarithms of primes. This is contrary to the major use of logarithms in multiplication/division! Therefore, in a sense, (2) reflects an "anomaly" or "paradox" about logarithms, which I will not further argue except to indicate that both the exponential of (2) and its additive/subtractive nature are arguably more "fundamental" than logarithmic or multiplicative/divisive natures as usually understood.

If we regard ln(pi) as some function f(pi) for i = 1 tp n, then (2) says:

3) I = exp[f(p1) + f(p2) + ... + f(pn)]

Here the "logarithmic" nature of f(pi), although "hidden", also becomes more intuitively obvious as less critical per the above discussion, while the exponential function remains important.

Compare this with my recent postings on primes on two or three other threads here, and a pattern seems to emerge.

Osher Doctorow

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.