Re: JSH:Understanding constant terms

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 11/29/04 

Date: Mon, 29 Nov 2004 08:28:39 +0100

jstevh@msn.com (James Harris) writes:

> "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message news:<87llcmoxk6.fsf@phiwumbda.org>...
>> jstevh@msn.com (James Harris) writes:
>>
>> > rupertmccallum@yahoo.com (Rupert) wrote in message news:<d6af759.0411271657.2876eeae@posting.google.com>...
>> >>
>> >> Just because the factors are divisible by 7 when x=0, it doesn't
>> >> follow that the factors are divisible by 7 for all values of x.
>> >>
>> >> <snip>
>> >
>> > It follows from the distributive property.
>> >
>>
>> Care to step through that argument for us slowpokes?
>
> Sure.
>
> The factor g_1(x) has *two* parts, where one of them is the constant
> term, which is constant as it is in fact, 7, and 7 is constant.

Er, I guess. Trivially, we can write

  g_1(x) = (g_1(x) - g_1(0)) + g_1(0).
           ^^^^^^^^^^^^^^^^^ ^^^^^^ so-called constant bit
           ^^^^^^^^^^^^^^^^^ evidently the "non-constant" bit
>
> The other varies as x varies.
>
> Now the constant term goes from 7 to 1, which means that it is divided
> by 7.
>
> Understand?
>
> Well then, by the distributive property, the other term must be
> divided by 7 as well, as if you have two parts, then you can't get to
> one without going through the other.

What the distributive property says is (a + b)/c = a/c + b/c. You
have g_1(0)/7 is an algebraic integer (or something). You want to
conclude that for all x, g_1(x)/7 is an algebraic integer. All the
distributive property seems to give you is:

  g_1(0) / 7 = (g_1(0) / 7 - g_1(0) / 7) + g_1(0) / 7

or, if you prefer,

  g_1(x) / 7 = (g_1(x) / 7 - g_1(0) / 7) + g_1(0) / 7

Neither of these equations proves that g_1(x) is divisible by 7.

I don't see how I go from that to the claim that 7 divides
g_1(x) - g_1(0) for all x.

Maybe a little more detail? I still don't get it. 

-- 
I don't want to wine and dine and date you once or twice.
I want to hold you now.  I just want to spend the night.
You tell me a better plan.  Baby, I'm not a patient man.
    -- Jimmy Lafave, the romantic troubadour.

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