Problems & Puzzles: Puzzles

Puzzle 169. The unit and the sum of three signed prime-cubes

In one page of the Eric Weisstein's Mathworld Encyclopedia we may read that:

"Mahler proved that 1 has infinitely-many representations as 3 signed cubes"

The earliest* prime solution that I have obtained is:

697093 - 565033  - 541013 = 1

Question 1: Can you find the next  five solutions?

Sunday addition: at the risk of seeming crazy I would like to post one more question:

Question 2: Can you find a solution where two of the cubes are consecutive primes?

________
* earliest here means the minimal sum of the non-signed primes

Solution:

Jean-Charles Mayrignac sent (25/2/02) the following methodical references for speeding up this puzzle:

If you want to solve the problem, you can use the following documentation: http://euler.free.fr/docs/ get the files jap1.gif, jap2.gif, jap3.gif, jap4.gif, jap5.gif and jap6.gif. They expose how the problem can be solved using a sieving method.

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