42 is the answer

[Mostowski Collapse]

Rumors are, that finding an integer solution
took a lot of computing resources:

x^3 + y^3 + 42 = z^3

Can we nevertheless solve this on a PC in
less than 5 minutes?

[Nasser M. Abbasi]

Did you mean

x^3 + y^3 + z^3 = 42?

Given that it says at
<https://interestingengineering.com/science/the-sum-of-three-cubes-problem-for-42-has-just-been-solved>

"Charity Engine uses a computer's idle processing power, and it took 1 million
hours of processing time to solve the Diophantine Equation where
k is equal to 42. The answer is:
(-80538738812075974)^3 + (80435758145817515)^3 + (12602123297335631)^3 = 42."

I do not see how this can be found in 5 minutes on a PC? No matter
how fast the PC is. They seem to have done brute force search to find this.
But may be they used improved search method as given in

<https://news.mit.edu/2021/solution-3-sum-cubes-puzzle-0311>
under "A solution’s twist" section. But even that will stil require
very large brute search method.

But I tried

eq=x^3+y^3+42==z^3
FindInstance[eq&&Element[{x,y,z},Integers],{x,y,z},1]

And it said

FindInstance::nsmet: The methods available to FindInstance are insufficient to
find the requested instances or prove they do not exist.

in 0.001 seconds. I guess Mathematica still does not know about
the method used to solve for 42.

--Nasser

[Mostowski Collapse]

Well I mean:

x^3 + y^3 + 42 = z^3

You only need to consider positive x,y,z then.
Assume there is some ingenious approach to
speed it up, maybe could then also solve:

114, 390, 627, 633, 732, 921, 975.

[Mild Shock]

I can run the simpler:

x^3 + y^3 + 9 = z^3

In a web browser with Dogelog Player:

Example 71: Diophantine Modular
X = 216, Y = 52, Z = 217;
X = 52, Y = 216, Z = 217;
fail.
% Zeit 3574 ms, GC 7 ms, Lips 1696084, Uhr 01.09.2023 20:56 true.
https://jsfiddle.net/Jean_Luc_Picard_2021/d2njehtp

Since it is a different algorithm than the usual Prolog
CLP(FD), its faster than SWI-Prolog and Scryer Prolog,
even faster than desktop.

Have Fun!

P.S.: Unfortunately I couldn't cross yet the 42 barrier.

[Mild Shock]

What speed-up could a N-qubit quantum computer give
to this problem, find a positive integer tripple (x,y,z) such that:

x^3 + y^3 + 42 = z^3

[Mild Shock]

Lets say you don't trust what a quantum computer
finds as a tripple for (x,y,z) to satisfy:

x^3 + y^3 + 42 = z^3

You could then view it as a candidate triple
only, and still feed it into a normal computer

and see whether it is a solution. Kind of combining
approximate search with exact verification.