First, let me buy you a clue: real Mathematicians don't go around giving attribution every time they write down an equation somebody proved 200 years ago. Giving attribution the way you do is the signature characteristic of someone trying to impress people who don't know better.
Second, I have no interest in "bedazzling" you; in fact, I wasn't even thinking about you at all when I wrote it. I was thinking about the lurkers, and used it to illustrate my point, which was that e, i, pi, -1, and 0 are all important numbers in Mathematics, and are inter-related by the relation I posted. This is in contrast to your precious "phi" which notably is absent form Euler's relation.
Another interesting relationship that Euler discovered was Euler's phi function. [snip]
Another nice treatise on Euler's phi-function is provided at Euler phi-function[emphasis added]
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances." You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?
No doubt you will protest that you never said it was.... which is true; but then the question becomes why would you post it in the first place unless you thought it was related to the issue at hand?
Euler's phi (or totient) function of a positive integer n is the number of integers in {1,2,3,...,n} which are relatively prime to n.You:
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances." You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?He's dead, Jim!
I mean, let's all go to see Signs and have a few beers (I mean a lot of beers) and then go to some farm and make a cool looking crop circles.
It sure would be a lot better than this pointless bickering that is going on.
I find the subject of crop circles to be very interesting.
It would be better to have a civilized and polite discussion of the topic rather than the current flame war that is going on.
Well I do believe the Pythagorian Theorum still attributes to Pythagoras, and the Bernoulli Principle still attributes to Bernoulli. Fourier transforms still attribute to Fourier, and Maxwell's Equations still attribute to Maxwell. So don't try to be deceptive here, Dr. longshadow..
Giving attribution the way you do is the signature characteristic of someone trying to impress people who don't know better.
You must have forgotten about your own long list of attributations, such as in your comment, "and serious Mathematicians, like Peano, Cantor, Hilbert, Gödel, Whitehead, and Russell, didn't, and don't, spend their time working on it or worrying about what the value of "phi" is to a bazzillionth decimal place.
This is in contrast to your precious "phi" which notably is absent form(sic) Euler's relation.
Just because Euler didn't relate it in one of his famous formulas doesn't mean the number isn't important.
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances."
If you're trying to say that Euler's phi (or totient) function isn't the same as golden ratio phi, well no kidding Sherlock. PS: Will you explain to me why you keep trying to argue about "vibrational resonances"? Do you want to argue about oscillations at the atomic level, or resonance in general. I can't see what it has to do with anything I've said however...
You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?
Duh, yep.
No doubt you will protest that you never said it was.... which is true; but then the question becomes why would you post it in the first place unless you thought it was related to the issue at hand?
Because there IS a relationship between the golden phi and the Euler phi function... :)
From On Approximate Harmonic Division of n by phi(n)
A Problem Proposal
Let phi(n) denote Euler's totient function phi(n), giving the number of natural numbers less than n and relatively prime to n. By number in this discussion, I mean natural number.
Consider the numbers n divided by phi(n) in approximately the golden ratio, i.e., numbers n minimizing |(k / EulerPhi(k)) - golden ratio phi|, where the expression ranges over all k with some fixed number of digits. In other words, for some r, n / EulerPhi(n) is a best approximation to phi, where n must be selected from the set of r-digit numbers. Equivalently, n is a r-digit minimizer of |(k / EulerPhi(k)) - phi|. Here, golden ratio phi = (1 + sqrt(5)) / 2. Written in increasing order, these numbers n, which I call the harmonious numbers, determine a sequence whose first few terms are
3, 9, 39, 117, 351, 507, 3417, 10251, 30753, 58089, 92259
(Note: These appear as sequence A065657 in the Online Encyclopedia of Integer Sequences by N. J. A. Sloane.)
For example, |(3 / EulerPhi(3)) - phi| = .118034 (approximately) is minimal for all one-digit numbers. |(117 / EulerPhi(117)) - phi| = .006966 (approximately) is minimal for all three-digit numbers.
Of course, n / phi(n) can only approximate the golden ratio, which is irrational.
Problems
- Can |(n / EulerPhi(n)) - golden ratio phi| be made arbitrarily close to 0? If yes, then find a function relating the accuracy epsilon to the number of digits of n.
- The listed terms have this property: for r = 1,...,5, all r-digit terms share the same set of prime factors. For example, all 4-digit terms have prime factors 3, 17, 67. Furthermore, all listed terms are multiples of 3. I conjecture that these hold in general.
Similar questions can be asked for the sequence of numbers n divided by n - phi(n) in approximately the golden ratio. (Consider instead the expression |(k / (k - EulerPhi(k))) - golden ratio phi|).
The first few terms of this sequence (the dual harmonious numbers) are
6, 10, 20, 40, 50, 80, 230, 460, 920, 5278, 10556, 21112, 36946, 42224, 68614, 73892, 84448
(Note: EIS Sequence A065758) Are all terms multiples of 2?
A Related Optimization
Consider the numbers n which, for some r, are r-digit maximizers of n/EulerPhi(n). The first few terms of this sequence are
6, 30, 60, 90, 210, 420, 630, 840, 2310, 4620, 6930, 9240, 25410, 50820, 76230.
(Note: EIS Sequence A065800) It is not hard to show that for r > 1, the first r-digit term of the sequence is the smallest r-digit primorial, if it exists. It remains to investigate the first terms when existence fails. It is also not hard to see that for r > 1, the r-digit terms are in arithmetic progression with common difference equal to the smallest r-digit term. For example, 210, 420, 630, 840 are in arithmetic progression with common difference 210. Obviously the r-digit minimizer of n/EulerPhi(n) is the largest prime of n digits.
I invite readers to communicate their solutions or comments by contacting me at the email address below. I will report any progress on these problems in this web page, and of course, acknowledge correct solutions.
Joseph L. Pe
iDEN System Engineering Tools and Statistics
Motorola Center
Schaumburg, IL
email:josephpe@excite.com