The Sum of Three Cubes Problem For 42 Has Just Been Solved

Using crowd sourced computing power, a University of Bristol mathematician has solved the last Diophantine Equation, the number 42.

Back in April, 2019, we wrote about the "Sum of Three Cubes" problem having been solved for the number 33. The problem can be stated as:
k = x3 + y3 + z3, where k is a whole number.
For example,
29 = 33 + 13 + 13
26 = 114,844,3653 + 110,902,3013 + –142,254,8403.


Equations of this form, k = x3 + y3 + z3, where k is between 1 and 100 are called Diophantine Equations, and are named after the Greek mathematician Diophantus of Alexandria who lived around 250 AD.

The 1950s yields solutions

Starting in the 1950s, mathematicians began working on solving Diophantine equations, and they found solutions for all the numbers except 33 and 42.

The numbers 4, 5, 13, 14, 22, 23, 31, 32 can never be expressed as the sum of three cubes because they can be written as:
9 x k + 4 or
9 x k + 5, where k is any integer.

That's where things stood until April 2019, when Andrew Booker of the University of Bristol created a new computer algorithm to solve the sum of three cubes problem for the number 33 that examined the number line in both directions all the way up to 99 quadrillion. That's 99,000,000,000,000,000.

The number line
The number line Source: Marcia Wendorf

The answer Booker found for 33 is:
88661289752875283 + -87784054428622393 + -27361114688070403
But, 42 remained unsolved.

The mystery of 42

Lewis Carroll, author of the Alice in Wonderland books was actually a mathematician, and he was obsessed with the number 42. There are 42 illustrations in Alice's Adventures in Wonderland. "Rule Forty-two" in that same work states, "All persons more than a mile high to leave the court," and in The Hunting of the Snark, the Baker has "forty-two boxes, all carefully packed, With his name painted clearly on each."

Fast forward 114 years, and English author Douglas Adams publishes his wildly popular book, The Hitchhiker's Guide to the Galaxy. In it, a supercomputer named Deep Thought spends 7.5 million years pondering the ultimate question, unfortunately once it finds an answer, no one can remember what the question was. The answer, however, is "42".

So, what is it with the number 42? First, it is the sum of the first 6 positive even numbers:
2 + 4 + 6 + 8 + 10 + 12 = 42.

42 is a pronic number, which are numbers that are the product of two consecutive integers:
n(n + 1)
2 x 3 x 7 = 42.

42 is also an abundant number, also called an excessive number. These are numbers where the sum of their proper divisors is greater than the number itself. Divisors of 42 are:
1, 2, 3, 6, 7, 14, and 21, which add up to 54.

42 is also a sphenic number, which are positive integers that are the product of three distinct prime numbers. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller numbers. In the case of 42, it is the product of:
2 x 3 x 7 = 42, where 2, 3 and 7 are prime numbers. 

The magic of 42

You can create a magic square whose sum is 42:

42 Magic Square
42 Magic Square Source: Marcia Wendorf

And, there is a magic cube whose vertices add up to 42:

42 Magic Cube
42 Magic Cube Source: Jaksmata/Wikimedia Commons


The fall of 42

Andrew Booker knew he was going to have to go greater than 99 quadrillion to find a solution to 42, and he teamed up with Massachusetts Institute of Technology mathematician Andrew Sutherland, who hooked him up with Charity Engine. Charity Engine is a crowdsourced "worldwide computer" comprised of some 500,000 home computers around the world.

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.

If you're thinking that the question of the meaning of life has been answered, ponder this: between the numbers 100 and 1,000, the sum of three cubes problem hasn't been solved for the numbers:
114, 165, 390, 579, 627, 633, 732, 906, 921 and 975.
Back to the drawing board.

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