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Take Another Look at p1p2...pn = exp[ln(p1) + ln(p2) + ... + ln(pn)]
Posted:
Mar 9, 2009 11:59 PM
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>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 = p1p2...pn, 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
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