Pythagoras meets Ramanujan

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• 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.

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
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