Problems & Puzzles: Puzzles

Puzzle 386. CabTaxi, prime version

Christian Boyer sent the following ´puzzle

The prime version of the Cabtaxi problem (the smallest number expressible as the sum OR DIFFERENCE of two [prime] cubes in K different ways).

After the prime version of the Taxicab problem (see Puzzle 90), here is the prime version of the Cabtaxi problem.

Here are "standard" Cabtaxi numbers, using cubes of prime or non-prime numbers:
in two ways: 91 = 6^3 - 5^3 = 3^3 + 4^3
in three ways: 728 = 6^3 + 8^3 = 9^3 - 1^3 = 12^3 - 10^3

You can read more details on Cabtaxi numbers here:
http://mathworld.wolfram.com/CabtaxiNumber.html
http://euler.free.fr/taxicab.htm
and very latest results at www.christianboyer.com/taxicab 

If we allow only cubes of PRIMES, the smallest solutions in two ways are far bigger than 91. If I am right, they are:

62540982 = 397^3 - 31^3 = 1867^3 - 1861^3
105161238 = 193^3 + 461^3 = 709^3 - 631^3
258428648 = 193^3 + 631^3 = 709^3 - 461^3 (a different arrangement of the previous solution)
349211772 = 461^3 + 631^3 = 709^3 - 193^3 (the other possible arrangement)
544861170 = 599^3 + 691^3 = 1033^3 - 823^3
...

We remark that 62540982 is a different arrangement of 6507811154 listed in Puzzle 90.

It seems that nobody has a solution of the question 1 of Puzzle 90: maybe the code of Alberto Hernandez is still running, since May 2000 ;-)

But because Cabtaxi numbers are smaller than Taxicab numbers, (numbers which are sums or differences of cubes are of course a large superset of numbers which are only sums of cubes), maybe somebody will find an answer to the similar new question 1 below: should be "easier" than question 1 of Puzzle 90, but after some tests, no solution for any N < 10^12... (astonishing
because as seen above, if non-prime numbers are authorized, N = 728 is the
easy solution for K=3).


Questions:

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

2. If you can't solve question 1 and get a 3-way solution using 6 cubes of primes, you can try to use less primes. Here is a 3-way solution using 4 cubes of primes (17^3, 41^3, 2^3, 89^3):
68913 = 40^3 + 17^3 = 41^3 - 2^3 = 89^3 - 86^3
Can you get another 3-way solution using at least 4 cubes of primes on
the 6 cubes?

 
 

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.

***

On Apr 16, 2008 Christian Boyer wrote:

I am pleased to inform you the first known solutions of my question 2.
Uwe Hollerbach, USA, is currently doing an exhaustive search, looking if my upper bound 933528127886302221000 is the true Cabtaxi(10) number.
As announced in www.christianboyer.com/taxicab, he found during this search:

439926932718 = 76393 - 18013 = 90073 - 66253 = 112573 - 99553 (second smallest solution using 4 primes)

3607745483160 = 153493 - 20293 = 183133 - 136333 = 293343 - 278643 (third smallest solution using 4 primes)

7070014626807314 = 1595353 + 1443793 = 1988173 - 923993 = 2287573 - 1698593 (smallest solution using 5 primes)

104886396577678268754 = 46858873 + 12590513 = 48737633 - 22160573 = 58573093- 45799153 (another solution using 5 primes)

But (very difficult) question 1 still unsolved: during his search in progress, maybe Uwe will also find the first 3-way solution using 6 primes?

***

J. Wroblewski wrote (May 08):

I have found the trhird 5-prime solution:

119375213^3 - 74413727^3 = 108883871^3 - 12172313^3 = 102944713^3 + 58296773^3

with 12172313 being composite.

Although my search was not exhaustive, I am getting an impression that
solutions with 5 primes are very rare, and existence of a perfect
solution involving 6 primes in a computable range is not very likely.

You might ask me: "Do you think a 6-prime solution exists at all?"
Well, I do not have any grounds for an educated opinion on that
question. However my intuition tells me that the answer YES is
somewhat more likely than NO.

Later he added

I have found the trhird 5-prime solution:

119375213^3 - 74413727^3 = 108883871^3 - 12172313^3 = 102944713^3 + 58296773^3

with 12172313 being composite.

Although my search was not exhaustive, I am getting an impression that
solutions with 5 primes are very rare, and existence of a perfect
solution involving 6 primes in a computable range is not very likely.

You might ask me: "Do you think a 6-prime solution exists at all?"
Well, I do not have any grounds for an educated opinion on that
question. However my intuition tells me that the answer YES is
somewhat more likely than NO.

Soon (May 12) C. Boyer added:

Another 5-prime solution found yesterday by Uwe Hollerbach, added at
www.christianboyer.com/taxicab We know now four 5-prime solutions, but still no 6-prime solution.

One day after:

Still another solution using 5 primes found by Uwe, added in
www.christianboyer.com/taxicab  Our congruence remarks are no longer valid, it was a coincidence: the last digit of this new solution is a 6.

***

 

 

 

 
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