Cosmology

[YesNo]

What I don't understand is what it means to "reduce to a pure factor of 2". The totient could only have one factor of 2 in it. Let n be a prime of the form 4m+3 to get a totient with only one factor of 2 in it.

[YesNo]

Regard Lehmer's conjecture, I can see why any such n so that φ(n) | n - 1 must be a Carmichael number. The order of any prime p dividing n would divide φ(n) and so divide n - 1 implying that p[sup]n-1[\sup] = 1 (mod n): http://mathworld.wolfram.com/LehmersTotientProblem.html

What about the other results?

(1) Why must it be square-free? (Edit: I can see why it must be square-free. If p^r divides n then p^r-1(p-1) divides n - 1. So p divides both n and n - 1 if r > 1. That implies p divides 1. So it must be square-free.)

(2) Why must it have at least 7 (or 11 or 14) distinct primes dividing it?

(3) Why must it be greater than 10²³?

(4) Why must it be odd? (Edit: I can see why it must be odd. If n were even then n - 1 is odd. By assumption φ(n) | n - 1, but φ(n) is even for n > 2. Considering the two cases for n <= 2: For n = 2, n is prime and since the conjecture only applies to composite n, n = 2 does not count. Since 1 is not a composite number either, n = 1 is not covered by the conjecture either.)

Here's another reference: http://math.stackexchange.com/questi...otient-problem

(5) Why are there no counterexamples of the form k2^k + 1 (Here's the paper: http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf)

(6) Why are there no Fibonacci numbers as counterexamples?

(7) Is the following solution to the problem correct or not? https://www.youtube.com/watch?v=swbZqPjrcGk (Edit: This doesn't convince me. We can skip the powers of b since we know n must be square-free. So let composite n = bC where b is prime then if φ(n) = n - 1, we have φ(bC) = φ(b)φ(C) = n - 1 = bC - 1. So φ(C) = (bC - 1)/(b - 1). I don't see why that could not reduce to an integer. It would be an integer if C = 1 (more generally C = b^k), but then n would be prime or not be square-free.)

I have skimmed through Lehmer's original paper, "On Euler's Totient Function": https://projecteuclid.org/download/p...ams/1183496203

There are many results in this paper. Bouris does not appear to have proved his result and the technique used of showing that a ratio cannot be an integer is used in Lehmer's paper. So I will skip (7).

[desiresjab]

What I don't understand is what it means to "reduce to a pure factor of 2". The totient could only have one factor of 2 in it. Let n be a prime of the form 4m+3 to get a totient with only one factor of 2 in it.

Let me do a better job of being clear. Sorry for the confusion.

Iterates to a pure power of 2.

[desiresjab]

I have skimmed through Lehmer's original paper, "On Euler's Totient Function": https://projecteuclid.org/download/p...ams/1183496203

There are many results in this paper. Bouris does not appear to have proved his result and the technique used of showing that a ratio cannot be an integer is used in Lehmer's paper. So I will skip (7).

I looked at the Bouris paper. One pass was really not enough, slow as I am. Almost every time he states that a proposition "assumes" something (at least fifteen times, it seemed) it would be necessary for me to think long and hard to verify his contention. I did not feel it was worth it, especially after glancing at the YouTube side menu where it seemed Bouris might have had other proofs of many famous propositions. He is obviously more than a crackpot, but I cannot go about verifying or unverifying every proof someone claims to have made of a famous proposition. The fact that he made this one in language I understand means I could follow it out, if I felt it was worth a prodigious effort. It was good mental exercise. To follow every detail completely would be too much exercise. A year from now I may look at a proof like this and follow it easily if past is precedent. I envy you if you can. But for now I will rely on your opinion of Bouris and marshall my strength for whatever takes me. I hear Liouville calling, yet I don't know. I also hear triangles calling, Euler calling, theories of categories and forms calling, class numbers calling...The great part about being retired and a math butterfly is that where I go is usually based on inspiration or the need to fulfill other inspirations.

I look at all links provided. Sometimes I have already read it. I often re-read things many times. If I find something I like, I will stay with it for days until I have drained it as well as I can.

One could spend forever looking at the connections of the Euler ф function--it is that centrally placed. One could find an involvement for it in practically any proposition. In short, what it does in number theory is stand in for the term p-1 in case of composite numbers. The rest of the time it is p-1. Armed with this idea and a few cogent connections, one may be able to go big game hunting in the wild and have a reasonable chance of spotting the beast in camouflage.

[YesNo]

I don't think Bouris has a solution. When I was searching for information on the problem his paper kept popping up so I had to consider it.

However, the Lehmer paper is worth reading. It contains the main ideas and proof techniques.

I found an old book on continued fractions by C. D. Olds that I started reading. This should help build a foundation for Liouville numbers.

[desiresjab]

I looked at the Lehmer paper too. I do not claim to have understood every ounce, though stylistically it was so much cleaner and easier than the Bouris to follow. For me it was a confirmation of my beliefs concerning the importance of style in mathematics. It is impossible to always be clear for those with less understanding of a topic, but clarity carries great weight as far as it can go, especially for readers. Lehmer's style takes this into account, that of Bouris did not seem to, at least for me it did not. A number theorist who became a master at explaining abstract concepts was H.L. Davenport.

[desiresjab]

I don't think Bouris has a solution. When I was searching for information on the problem his paper kept popping up so I had to consider it.

However, the Lehmer paper is worth reading. It contains the main ideas and proof techniques.

I found an old book on continued fractions by C. D. Olds that I started reading. This should help build a foundation for Liouville numbers.

Continued fractions are plumb scary!

[desiresjab]

In the meantime, though, I may be looking at decimal expansions, a topic clearly related to continured fractions.

[desiresjab]

I figured I would run into the DeuceHound formula, if I kept reading. See a close variant in the aritcle below used in a product. What is the formula for? The sum of the divisors of n, of course, usually denoted σ(n), proving I have not yet been around in the basic number theoretic functions as much as I need to be, or I would have recognized this. I will never forget it. The divisors functions (along with a few others), I have bascially ignored, but I have seen in the last few days that they are terribly well connected.

https://mathlesstraveled.com/2007/11...umbers-part-i/

[YesNo]

I liked how he put the latex math symbols on that wordpress page.

[desiresjab]

I liked how he put the latex math symbols on that wordpress page.

What?

[YesNo]

I didn't know Wordpress sites could format mathematics formulas the way that site formatted them.

[desiresjab]

I didn't know Wordpress sites could format mathematics formulas the way that site formatted them.

I have never even heard of WordPress.

[Dreamwoven]

Wordpress have their own blogs, I have wordpress blogs and they allow public re-blogging of posts, which is very handy. See this: https://wordpress.org/news/

[desiresjab]

Wordpress have their own blogs, I have wordpress blogs and they allow public re-blogging of posts, which is very handy. See this: https://wordpress.org/news/

How much do they want for it?