Sorry, for replying so late. Start with the set S = {0, 1, 6, 11, 16, 21}. As you observe, squaring doesn't fix some of the elements, such as 6, for which 6^2 == 11 (mod 25). However, if you square all of the elements, you get {0, 1, 11, 21, 6, 16}, which is the same set. So the set S is fixed by squaring, even though some if its elements are not.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of William R Somsky Sent: Thursday, November 10, 2016 8:54 PM To: math-fun Cc: Dan Asimov Subject: Re: [math-fun] Mapping problem
I don't get it. Why is 6 in S(25)? 6^2 = 36 = 11 mod 25, not 6. And if it's because 11 I'd in the list too, then the set oh all 0 .. 24 would work as well...
On Nov 8, 2016 23:10, "David Wilson" <davidwwilson@comcast.net> wrote:
By Z(n) I meant the ring of integers modulo n, perhaps Z/n or Z/nZ is better?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Wednesday, November 09, 2016 12:43 AM To: math-fun Subject: Re: [math-fun] Mapping problem
I'm not feeling particularly patient at the moment.
Can you kindly say what the notation "Z(n)" stands for?
I can try to guess, of course, but I would rather not.
—Dan
----- Let S(n) be the largest subset of Z(n) fixed by the mapping n -> n^2, and let f(n) = |Z(n)|. For example, S(25) = {0, 1, 6, 11, 16, 21} is the largest set of residues modulo 25 fixed by the mapping n -> n^2, so f(25) = |S(25)| = 6. Can you find a formula for f(n) in terms of n? -----
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