[seqfan] Re: number-theoretic notation

[Olivier Gerard]

On Tue, Sep 11, 2012 at 12:39 PM, Ed Jeffery <lejeffery7 at gmail.com> wrote:
> Olivier,
>
> Wow, I sure thank you for that: I had no idea what it meant, and my eyes
> hurt from searching. Now I am curious again: what does your notation =!=
> mean; is that something from Mathematica syntax?
>

I was just rephrasing Jean-Pierre explanation on the usual difference
 between p|m and p||m,
because I guessed you did not catch it fully.

About my =!= notation, it is just an ASCII
way to write not equal (incongruent)

It means only m   is not equal to    0 modulo p^2

If you want to use these ideas with Mathematica, lookup the
IntegerExponent function. For instance these three programs give the
same result

Select[Range[1000] , IntegerExponent[#, 3]==2 &]
Select[Range[1000] , Mod[#,3^2]==0 && Mod[#,3^3]>0 &]
Select[Range[1000] , MemberQ[FactorInteger[#], {3, 2}] &]

i.e. numbers less than 1000 which are exactly divisible by 9,
but the first one is more efficient and elegant.

Olivier


>>Just a warning
>
>>p || m    is not equivalent   to m == 0 (mod p).
>>but to the conjunction  (m == 0 (mod p)  AND m =!= 0 (mod p^2))
>
>>like wise...
>
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