From: fredh@ix.netcom.com (Fred W. Helenius) Newsgroups: sci.math Subject: Re: FLT ?Generalized? (I hope not!) Date: Tue, 05 Nov 1996 20:58:41 GMT grandenj@thegrotto.com (Jesse Granden) wrote: >I have a question that I would like answered. a couple of years ago while I >was in high school, my math teacher introduced me to FLT. This made me >generalize the following: >a1^n + a2^n + a3^n. . .am^n = z^n has no soultions for n>m ><-----m terms-------------> >where m, n, a1 a2...am are positive integers This generalization is known as Euler's conjecture, and it is false. In 1966, Lander and Parkin found that 27^5 + 84^5 + 110^5 + 133^5 = 144^5. More recently, the fourth power case was settled by Noam Elkies; a smaller example than his, due to Roger Frye, is 95800^4 + 217519^4 + 414560^4 = 422481^4. As far as I know, it is still not known whether a sixth power can be expressed as the sum of _six_ sixth powers, much less whether it can be done with five. Higher powers are also still open. -- Fred W. Helenius ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Where to look for latest work re. proof of a^n + b^n + c^n = d^n (0 < n < ?) Date: 6 Jan 1998 06:26:09 GMT In article <01bd14a2$35c4f680$140010ac@Developer2.admin.intouchsurvey.com>, Russell Harper wrote: >Can anyone give me some pointers as to where I could look for the most >recent work in the following problems? > >Given a^n + b^n + c^n = d^n where a, b, c, d, n are all positive integers. > >1) Prove or disprove an upper bound for n. Open. If I had to guess, I'd probably say there are solutions iff n <= 4. >2) If a solution exists for a particular n, then do solutions exist for all >integers m where 0 < m < n? I see no reason why that need be true. >In a similar posting I sent some time ago, I received numerous examples for >n = 4. There are many easy solutions for 0 < n < 4. For n > 4, there >doesn't appear to be too much work on the subject. Au contraire; perhaps much work and few results. What _has_ been done is to put the problem into a number of more general settings, in which one can hope to make some sense of a Big Picture. For example, the equation you propose defines an algebraic surface; rather than focus on the one equation, it is more rewarding to ask, what can one say in general about algebraic surfaces -- what kind of structure do they have? what can we measure about them? how different from one another can they be? As I remarked in another post, there is a web page which summarizes (or provides links to summaries of) quite a bit of what's known and what's not about equal sums of like powers. The URL is http://www.jiangmen.gd.cn/person/chen/eslp.htm dave