Problems & Puzzles: Puzzles Puzzle 90.
The prime version of the taxicab problem
(the smallest number expressible as the sum of two [prime] cubes in two
different ways). Let's
read once more this magnificent little story: "Once,
in the taxi from London [to Putney], Hardy noticed its number,
1729. He must have thought about it a little because he entered the room
where Ramanujan lay in bed and, with scarcely a hello, blurted out
his disappointment with it. It was, he declared, "rather a dull
number," adding that he hoped that wasn't a bad omen. "No,
Hardy," said Ramanujan. "It is a very interesting
number. It is the smallest number expressible as the sum of two [positive]
cubes in two different ways." (taken from http://www.lacim.uqam.ca/pi/problem.html) Last
week, after reading the above quote, I asked my self for the least N such
that N=a^3+b^3=c^3+d^3, being a, b, c & d all distinct prime numbers
(obviously N can not be prime) Here
are my results (primes in blue, as usual, composites in red) 6058655748
= 61^3 + 1823^3 = 1049^3 + 1699^3
(this is the least for K=2 ways) Other results, but not the least  of course  are (I'm omitting the cubic exponent): 6507811154
= 31 + 1867 = 397 + 1861 Questions:
Solution Alberto Hernández, from Monterrey Nuevo León, found (15/05/2000) the following least solutions for questions 2 & 3: 2) 1799027 = 13^3+83^3+107^3=11^3+89^3+103^3 3) 7231013=13^3+137^3+167^3=17^3+127^3+173^3 His code is running after the first solutions to question 1... Good luck to him!... *** On Mar 25, 2008 Christian Boyer wrote:
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