From: "Robert Harrison"
Newsgroups: alt.fan.hofstadter,alt.tanaka-tomoyuki,sci.math
Subject: Re: Fermat, Goldbach, open problems
Date: Wed, 23 Dec 1998 19:59:27 -0000
steiner wrote in message ...
>In article <75mtnv$qk9$1@mark.ucdavis.edu>, ez074520@dilbert.ucdavis.edu
>(Tomoyuki Tanaka) wrote:
>
>>
>> i just read:
>> "Fermat's Enigma : The Epic Quest to Solve the World's
>> Greatest Mathematical Problem"
>> by Simon Singh
>>
>> is there a book that has a bit more on modular forms and
>> elliptic curves, that is _not_ a real math book?
>>
>> i wondered if there are many open problems that can be easily
>> described like the Goldbach Conjecture. (not necessarily
>> number theory.)
>>
>>--------------------------------------------------------------------
>>http://www.astro.virginia.edu/~eww6n/math/GoldbachConjecture.html
>>
>> [...]
>> the ``strong'' Goldbach conjecture) asserts that all Positive
>> Even Integers >= 4 can be expressed as the Sum of two Primes.
>
>Two that come to mind are
>1). Catalan's conjecture: The only powers that differ by 1 are 9(= 3^2)
>and 8(= 2^3).
>2). The 3x+1 problem. This has often been called "The simplest unsolved
problem
>of arithmetic".
>There are many other examples in an old book "Tomorrow's Math", by C.S.
Ogilvy.
>(Wonder if a new edition has appeared since the late 1960's?)
>Regards,
>Ray Steiner
The two problems are related, if their exist numbers
s and t such that 2^t-3^s=1 then the 3x+1
(Collatz) problem is solved. The t will be the
number of even numbers in the cycle, the s the
number of odds. The 2^2-3^1 = 1 values for s and t
is related to the know 'attractor' of the Collatz
problem the 1->4->2->1 cycle.
Robert Harrison
==============================================================================
[Note that except for 2^1=2 and 2^2=4, all powers of 2 reduce to
{8, 16, 32, 48, 64}
mod 80, while the powers of three are
{1, 3, 9, 27}
so 2^t - 3^s requires t=1 or 2, and so 3^s = 1 or 3 (s=0 or 1,
respectively). That is, {2,1} and {4,3} are the only such pairs. -- djr]