TimeStamp: 07/13/97 at 02:19:01
From: Doctor Terrel
To: LMARTIN-NORTHFIELD@worldnet.att.net (LAWRENCE MARTIN)
Subject: Re: Goldbach's Conjecture
Approved-by: sydney
Date of most recent message in thread: 07/09/97 at 00:25:46

As LAWRENCE MARTIN wrote to Dr. Math
On 07/09/97 at 00:25:46 (Eastern Time),
>At 06:05 AM 7/8/97 +0000, you wrote:
>
>>As to your main question:  I'm not aware of any "proof" of
Goldbach's 
>>Conjecture existing yet.  (Maybe you'll be the one to find it
someday.  :) 
) 
>> But that doesn't mean it should be abandoned or ignored.  One thing
I 
often 
>>ask my students to do is to find all possible prime number pairs
whose sum 
>>is a particular even number, say 100.  For example, 3 & 97 make
a sum of 
>>100, but so do 11 & 89.  My question then becomes: how many more
pairs can 
>>be found?  It usually gets my students to thinking a lot, plus they
learn 
>>more about the primes less than 100.
>
>I read about that in the book GREAT MATHEMATICAL MYSTERIES.  It seems
that
>the higher the even number, the more ways there are of representing it as
>the sum of two primes. This makes a counterexample extremely unlikely,
but,
>as mathematicians know, that is no proof!
>
>>Perhaps you are not aware of another lesser known extension to the
main, 
>>popular Conjecture, the 3 prime case.  It says: All odd integers
greater 
>>than 5 can be expressed as the sum of THREE primes.  For example, 
>
>        I have heard of this. It is called Goldbach's Ternary Conjecture,
>while the better known part is Goldbach's Binary Conjecture.  This was
also
>in the book I referred to above.  It said that the Ternary Conjecture has
>been proven for every "sufficiently large" number.  Although the book 
didn't
>say this, I figured out that if Goldbach's Binary Conjecture is true,
>Goldbach's Ternary conjecture would have to be true.  However, if the
>Ternary Conjecture is proven true for sure, that would not automatically
>mean that the Binary Conjecture is true.
>        
>        Thanks for answering my letter!  Could you tell me if you know of
>any other good books besides the one I spoke of above, GREAT MATHMATICAL
>MYSTERIES, that contain good discussions of Goldbach's conjecture?
>
>                        Sincerely,
>                        Daniel
>
Dear Daniel,

Thanks for your reply to my answer of your original question.  I see you
are quite interested in Goldbach's Conjecture.  I liked your use of the
more formal names: Binary and Ternary for the 2- and 3-prime cases.

I would like to make one general observation, however, to a comment of
yours.  You said: "It seems that the higher the even number, the more
ways there are of representing it as the sum of two primes."  This is
not entirely true.  It is true that larger numbers will, in general,
have more representations than smaller numbers, but not always.  There
are many counterexamples I could give.  Here is just one: 96 has 7 prime
pairs, yet 98, which is larger, has only 4 prime pairs.  [The proof of
this is left as homework!  ;) ]

Question for you (and all other readers of this note):  Which number,
less than or equal to 100, has the greatest number of prime pairs? 
[HINT:  it is not 96.]

I highly commend your interest in this area of recreational
mathematics.  I myself enjoy the whole field very much.  While I do not
know of any book with more information about the Goldbach Conjecture, I
would like to recommend this book for your study and pleasure: 
MATHEMATICS ON VACATION, by Joseph S. Madachy.  Charles Scribner's Sons,
New York, 1996.  It contains many items about number oddities, similar
to what we are discussing.

Wishing you continued success,

-Doctor Terrel,  The Math Forum
 Check out our web site!  </dr.math/>

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