Loading ...
Sorry, an error occurred while loading the content.
 

Pythagoras meets Ramanujan

Expand Messages
  • Halliday, Ian
    As you will probably know, 1729 is the smallest number expressible as the sum of two cubes in two distinct ways. (a taxicab number) Similarly, Pythagoras
    Message 1 of 1 , Dec 29, 2001
      As you will probably know, 1729 is the smallest number expressible as
      the sum of two cubes in two distinct ways. (a taxicab number)

      Similarly, Pythagoras showed that there are numbers which are squares
      which are the sum of two squares.

      The current puzzles relate to equal sums of two squares:

      a^2 + b^2 = c^2 + d^2 = T,
      a > 0, b > 0, c > 0, d > 0,
      a != c, a != d, b != c, b!= d

      The questions are
      (1) with these conditions,
      what is the smallest value of T ?

      (2) add conditions a != b, c != d
      what is the smallest value of T ?

      (3) add conditions a > 1, b > 1, c > 1, d > 1
      what is the smallest value of T ?

      (4) add condition T not divisible by 5
      what is the smallest value of T ?

      (5) add condition T is prime (seeing as this is a prime number mailing
      list)
      what is the smallest value of T ?

      In each case, of course, showing that there is no such T suffices as a
      solution.
      Answers by private email please, summary posted around mid January.

      Have a happy new year.

      Regards,

      Ian
      --
      Ian W Halliday, BA Hons, MIMIS, AAIBF Snr, ATMB, CL
      +64 27 245 6089 (GMT+13)
      http://baptism.co.nz
      Focus On Success
      --
      Word documents not accepted -- see http://baptism.co.nz/word.html

      [Non-text portions of this message have been removed]
    Your message has been successfully submitted and would be delivered to recipients shortly.