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
12906787894 = 593 + 2333 = 1787 + 1931
20593712932 = 71 + 2741 = 977 + 2699

Questions:

1. Can you get the least N expressible as the sum of two prime cubes in K>2 ways, for K=3, 4 & 5?

2. Can you get the least N expressible as the sum of three prime cubes in 2 way.

3.  Can you get the least P (prime) expressible as the sum of three prime cubes in 2 ways

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:

My paper "New Upper Bounds for Taxicab and Cabtaxi Numbers" is now published in the Journal of Integer Sequences. Look at www.christianboyer.com/taxicab. In part 8.2 (and in the References), I was pleased to mention your puzzles 90 and 386.

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