### wordpress *won’t* let me publish my copy! quitting forever again soon i bet! dammit!!!!

[link of the hour:

number theory and physics.]

********************************************

Define f(n) = n – \phi(n).

Then f is the function that, from

CRS(n)={0,1,…,n-1}

(a Complete Residue System (mod n)),

counts the elements “a”

such that (a,n) \not= 1.

We can think of this as counting

the objects of the set

ZD(n):= CRS(n) \setdifference RRS(n):

f(n) = #ZD(n).

(Objects of ZD(n) are sometimes called

“zero divisors” (mod n).)

Now if x \in ZD(d), we have (x,d) \not= 1

and so (recall d|n) we get (x,n) \not= 1 as well:

x \in ZD(n). Thus ZD(d) \subset ZD(n).

But then f(d) \le f(n),

or, in other words,

n – \phi(n) \ge d – \phi(d).

This is *almost* what we want…

the “\ge” (“is greater than or equal to”)

must be “made strict”… we must

*eliminate* the possibility that

the two sides of our equation are equal.

So notice that

d \not== 0 (mod n)

whereas

d == 1 (mod d).

Thus [0] and [d] represent

*different* objects (equivalence classes)

in ZD(n) but represent the *same* object

in ZD(d). It follows that the “\subset”

relation found three paragraphs above

is *strict*; our result follows:

n – \phi(n) > d – \phi(d).

February 23, 2014 at 9:29 pm

PS

(If d|n and 0<d d – \phi(d).

)

[Let “\phi” denote “Euler’s Phi-function”

(the “totient” function)

defined by

\phi(n) = #RRS(n).

We’ve seen that \phi

“counts numbers prime to n”…

or, the same thing, counts elements

of a Reduced Residue System (mod n)

(these are the a \in CRS(n) such that

(a,n) = 1.)

]

Let d|n with 0<d d – \phi(d).

February 23, 2014 at 9:34 pm

right at the end

there’s some secret code

(hidden from my eyes) that causes

some evil software-demon to throw

away what should *follow* from here

(namely the rest of the post as written;

the bit that appears at the top level

of this blog *as* a post).

this is *maddening*.

this used to be so *easy*!

February 23, 2014 at 9:36 pm

original title: “that’s how cyclic groups roll”.

February 25, 2014 at 4:46 pm

error patched moments ago.

after, like, a whole day. what,

no bug reports?

February 28, 2014 at 5:20 am

https://people.math.osu.edu/sinnott.1/BSwDtalk.pdf

sinnott on swinnerton-dyer (slides from a talk…

unused there because of “technical difficulties”.