The Diophantine equation $P_1^3 + P_2^3 + P_3^3 = P_4^3$

Question

Consider the Diophantine equation

$$P_1^3 + P_2^3 + P_3^3 = P_4^3$$

Where $P_n$ are distinct odd primes.

What are the smallest solutions ?

Is there even a solution ? Or is there a reason no such solution exists ?

I looked at some parametric solutions for the diophantine but those did not allow all of the variables to be prime.

I was not able to find an easy argument with basic modular arithmetic or p-adic.

Notice the pythagorean triples, that are lower degree analogues, do not allow more than one prime ;

$$A^2 + B^2 = C^2$$

the solutions are positive integers of type $a(x^2 + y^2),a(2xy),a(x^2 - y^2)$ so never more than 1 prime.

At the same time, the degree 4 variant

$$x^4 + y^4 + z^4 = w^4$$

does have solutions that are all prime I believe. So I wonder about this cubic case.

Answer 1

Based upon this SymPy code, I would say that there probably are no solutions. However, we still haven't ruled out incredibly huge prime number solutions, that are not feasible with SymPy.

from sympy import *

N = 500

P = list(primerange(2, N))    # Must make list in case generator object is only good for a one-off iteration;
# We'll be looping over this 4 times, nested!

solns = set()

for p in P:
    for q in P:
        for r in P:
            for s in P:
                if len({p, q, r, s}) == 4:    # Distinctness requirement met; see {} notation in Python, usually means dict, but
                    # here it means construct a set() object.
                
                    if p ** 3 + q ** 3 + r ** 3 == s ** 3:
                        print((p, q, r, s))
                        solns.add((p, q, r, s))
                        
print(solns)

Will print set() (empty set), meaning no solutions $p,q,r,s \leq 500$.

So next, you could try a C++ program which will (on this particular problem) run about 1000 - 10,000 times faster. Or you can tackle the problem theoretically. I am not sure how they do that in practice, but I've seen existence results of another but related type before.

Hope that helps. I see you also have a prime number addiction. You're in a dark place now, but remember, there is light at the end of the tunnel. But first you'll have to see through a few of these prime number questions you have to the very depths of prime number hell. There you will find the reasons why they need the advanced Algebraic/Analytic Number Theory to even begin to approach such problems, while even those approaches fail! Good luck with your prime number "adventures".