Problems & Puzzles: Puzzles

Puzzle 48.- P³ = a³ + b³ + c³, {P, a, b, c} = primes

Highly stimulated by the Russo's question (See Puzzle 47), I asked myself for the solution to cube of primes such that all the three cubes to be added are primes also (this is permitted by the parity involved).

The least solution that I have gotten is:

709³ = 193³ + 461³ + 631³

On the other hand, the first solution for P=palprime and {a,b,c} = odds is:
929³ = 69³ + 447³ + 893³ .
And is not very hard to find other solutions of this kind but…
a) does exists a solution being P a palprime and {a,b,c} = primes?
b) does exists a solution being P a palindrome and {a,b,c} = primes?

Felice Russo has found two solutions to b):

505^3=59^3+163^3+499^3 ..........(4/5/99)
535^3=349^3+ 379^3+383^3 ........(15/5/99)

***

On 1/7/6, J. Wroblewski sent the following solutions:

I have created the file:
http://www.math.uni.wroc.pl/~jwr/eslp/prime3.zip

This file contains complete list of 15567
nontrivial solutions to
x^3+y^3+z^3+t^3=0
with x,y,z,t being signed primes other than +-19
and with absolute value up to Prime[720000]=10,894,817.
I have ruled out 19 for code efficiency, assuming it skips
very few solutions, if any.

Below is the complete list of answers to Puzzle 48, question a):

3353533^3 = 342071^3 + 1555639^3 + 3236743^3
7141417^3 = 2368979^3 + 5524903^3 + 5669863^3
7941497^3 = 414487^3 + 5418779^3 + 6990911^3
7985897^3 = 1612439^3 + 3497891^3 + 7732327^3
9332339^3 = 2413097^3 + 5254687^3 + 8678507^3
9700079^3 = 5007017^3 + 5246053^3 + 8630249^3

If there is any other solution below 10,000,000, it must contain a 19^3
term on the right.

Some other interesting things I have found:

28477^3 = 10781^3 + 11071^3 + 27361^3
28477^3 = 3739^3 + 17203^3 + 26183^3

33199^3 = 15187^3 + 24197^3 + 26647^3
33199^3 = 2833^3 + 19081^3 + 30941^3
33199^3 = -12227^3 + 26641^3 + 26921^3

***