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]