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From: rpb@aaec.OZ (Bob Backstrom)
Newsgroups: net.math
Subject: Re: A number theory problem
Message-ID: <482@aaec.OZ>
Date: Sun, 1-Sep-85 23:54:33 EDT
Article-I.D.: aaec.482
Posted: Sun Sep  1 23:54:33 1985
Date-Received: Fri, 6-Sep-85 04:20:28 EDT
References: <388@aero.ARPA>
Organization: Australian Atomic Energy Commission
Lines: 48

> 
> Most of you have probably heard the story of Ramanujan, who was riding
> in the cab with a friend. They were discussing his room number 1729,
> when his friend remarked that it was an uninteresting number.
> "Oh no" Ramanujan replied. "it is the smallest number that can be written
> as the sum of two cubes in two different ways".
>  
> My question is, what is the smallest number that can be written as the
> sum of two cubes in THREE different ways? Does one exist?
> 
>                                    Bill Sinclair (asbestos Willie)

[ I am posting this reply to the net after getting Unknown Host when
mailing to "sinclair@aero.ARPA" ]
 
I had looked at exactly this problem some years ago and found the
first four such solutions as follows:

                 3      3
 87,539,319 = 167  + 436
                 3      3
            = 228  + 423
                 3      3
            = 255  + 414
  
                 3      3
119,824,488 =  11  + 493
                 3      3
            =  90  + 492
                 3      3
            = 346  + 428
  
                 3      3
143,604,279 = 111  + 522
                 3      3
            = 359  + 460
                 3      3
            = 408  + 423
  
                 3      3
175,959,000 =  70  + 560
                 3      3
            = 198  + 552
                 3      3
            = 315  + 525

From Bob Backstrom at the Australian Atomic Energy Commission,
Sydney, New South Wales, Australia.