HTTP/2 301 server: nginx date: Tue, 30 Dec 2025 23:28:35 GMT content-type: application/rss+xml; charset=UTF-8 location: https://yiminge.wordpress.com/comments/feed/ x-hacker: Want root? Visit join.a8c.com/hacker and mention this header. host-header: WordPress.com vary: accept, content-type, cookie last-modified: Tue, 30 Dec 2025 23:28:35 GMT etag: "024234cd6e845a9049259837e20dedcd" x-redirect-by: WordPress x-ac: 4.bom _dca MISS alt-svc: h3=":443"; ma=86400 strict-transport-security: max-age=31536000 server-timing: a8c-cdn, dc;desc=bom, cache;desc=MISS;dur=388.0 HTTP/2 200 server: nginx date: Tue, 30 Dec 2025 23:28:36 GMT content-type: application/rss+xml; charset=UTF-8 vary: Accept-Encoding x-hacker: Want root? Visit join.a8c.com/hacker and mention this header. host-header: WordPress.com vary: accept, content-type, cookie last-modified: Tue, 30 Dec 2025 23:28:35 GMT etag: W/"024234cd6e845a9049259837e20dedcd" content-encoding: gzip x-ac: 4.bom _dca MISS alt-svc: h3=":443"; ma=86400 strict-transport-security: max-age=31536000 server-timing: a8c-cdn, dc;desc=bom, cache;desc=MISS;dur=367.0 https://yiminge.wordpress.com Tue, 10 May 2011 06:10:41 +0000 hourly 1 https://wordpress.com/ https://yiminge.wordpress.com/2009/09/26/automorphism-groups-of-simple-graphs/#comment-121 Tue, 10 May 2011 06:10:41 +0000 https://yiminge.wordpress.com/?p=164#comment-121 They are only isomorphic not identical groups.

]]> https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff/#comment-50 Mon, 29 Mar 2010 13:46:24 +0000 https://yiminge.wordpress.com/?p=59#comment-50 My name is Piter Jankovich. oOnly want to tell, that your blog is really cool
And want to ask you: is this blog your hobby?
P.S. Sorry for my bad english

]]> https://yiminge.wordpress.com/2009/06/09/the-sum-of-primitive-roots-of-unity/#comment-48 Tue, 21 Jul 2009 10:32:03 +0000 https://yiminge.wordpress.com/?p=156#comment-48 Good article of Mobius function.

]]> https://yiminge.wordpress.com/2009/06/09/the-sum-of-primitive-roots-of-unity/#comment-47 Thu, 16 Jul 2009 13:35:36 +0000 https://yiminge.wordpress.com/?p=156#comment-47 Nice.

Of course, the argument you give is very neat and probably the best. It is, however, possible to verify it directly of course.

If , then since $(k,n)=1 \Leftrightarrow (k+st,n)=1$ the sum is divisible by .

Finally (since is manifestly multiplicative), for any prime , .

So is indeed the Mobius function.

]]> https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff/#comment-45 Thu, 18 Jun 2009 18:28:01 +0000 https://yiminge.wordpress.com/?p=59#comment-45 There is a formula, ascribed to Hölder, that gives the number of (isomorphically distinct) groups of order n, if n is ‘quadratfrei’. The formula is given in the Mathworld article ‘Finite Group’ but I think there are a couple of errors in that version of it. What I think are correct versions are contained in an article ‘Counting Groups…’ by Conway, Dietrich and O’Brien, which is on the web, and in another article, ‘Groups of square-free order are scarce’, by Mays, which is in the Pacific Journal of Mathematics and also on the web. Mays’ article gives a simple worked example of Hölder’s formula (which he ascribes instead to one ‘Balash’) and also refers to the ‘if’ part of your theorem.

If the only (isomorphically distinct) group of order n is the cyclic group, then n is ‘quadratfrei’ and Hölder’s formula applies. Then I think it can probably be shown that Hölder’s formula can only be made to yield the value 1 if your condition, that gcd(n, phi(n)) = 1, holds. Conversely if your condition holds, n is ‘quadratfrei’ and according to Mays, Hölder’s formula yields the value 1 in this case.

The article by Conway et al. gives an outline of how Hölder’s formula can be arrived at. The two papers of Hölder referenced in the Mathworld article can be accessed through the University of Göttingen website but they look like heavy going to me, and on a quick scan I can’t see the formula given explicitly.

Happy problem solving. I look forward to seeing many more beautiful results on your blog. You are exceptionally talented.

]]> https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff/#comment-43 Wed, 17 Jun 2009 23:53:36 +0000 https://yiminge.wordpress.com/?p=59#comment-43 You’re quite right, I misunderstood the proposition. Thanks for clarifying it for me.

]]> https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff/#comment-42 Wed, 17 Jun 2009 15:39:13 +0000 https://yiminge.wordpress.com/?p=59#comment-42 I think you got the statement wrong. The proposition claims that if gcd(n,phi(n))=1 then there do not exist groups of order n that are not cyclic and conversely, if gcd(n,phi(n))>1 then there are some noncyclic groups. For n=4, there is also the Klein four-group which is not cyclic ;-).

]]> https://yiminge.wordpress.com/2009/04/04/the-probability-of-coprimality/#comment-41 Wed, 17 Jun 2009 13:31:19 +0000 https://yiminge.wordpress.com/?p=137#comment-41 There is a discussion of this and related results in Chapter 18 of Hardy and Wright’s ‘Number Theory’ (5th edition) which you may enjoy. This particular result is their Theorem 332.

]]> https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff/#comment-40 Wed, 17 Jun 2009 11:19:55 +0000 https://yiminge.wordpress.com/?p=59#comment-40 Perhaps I’m overlooking something, but isn’t it the case that there is exactly one cyclic group of every order? There is one cyclic group of order 4 and yet phi(4) = 2 and gcd(4,2) = 2, contradicting your proposition.

]]> https://yiminge.wordpress.com/2009/04/04/the-probability-of-coprimality/#comment-39 Sat, 18 Apr 2009 13:51:08 +0000 https://yiminge.wordpress.com/?p=137#comment-39 Another interesting theorem slightly related to this one is the following:
Let pt(n) denote the number of primitive pythagorean triples with hypotenuse less than, or equal to n.

Then

I can send you the proof if you want (4 or 5 pages long)

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