42 is the answer

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Mostowski Collapse

Sep 19, 2022, 5:51:16 PM9/19/22
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

Sep 19, 2022, 10:24:19 PM9/19/22
Did you mean

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

Given that it says at

"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

under "A solution’s twist" section. But even that will stil require
very large brute search method.

But I tried


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.


Mostowski Collapse

Sep 20, 2022, 3:37:53 AM9/20/22
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.
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