### Sum of Three Cubes

March 14, 2022

Most

number theory conjectures are easy to understand, but they are often very difficult to

prove. One example of this is

Fermat's Last Theorem, a conjecture that there are no three

positive integers,

**a**,

**b**, and

**c**, that satisfy the

equation
**a**^{n} + b^{n} = c^{n}

for any integer

**n** > 2. This conjecture is named after

mathematician,

Pierre de Fermat (1607-1665), who claimed to have an

unpublished proof around 1637. In retrospect, this claim is quite unlikely. The actual proof, in 1995 by

Andrew Wiles, is 129 pages long.

There are a few number theory conjectures that are both simple to state and simple to prove, one example of which is the conjecture that the

square of any

odd number is also odd.

Anecdotal evidence from some odd numbers such as 3 (3

^{2} = 9), 11 (11

^{2} = 121), and 314159 (314159

^{2} = 98695877281) is apparent, but that's not a proof. The actual proof is to create an odd number by adding one to an

even number,

**n = 2k + 1**. Then

**
n**^{2} = 4k^{2} + 4k + 1

n^{2} = 2(2k^{2} + 2k) + 1

so,

**n**^{2} is odd, since it's the sum of an even number plus one.

Diophantine equations are equations in which

integer solutions are required. These equations as named after the

3rd century mathematician,

Diophantus of Alexandria (c.284-c.298), who studied such equations and compiled them in his

book,

Arithmetica. Unlike later

mathematics books, Arithmetica was a compilation of problems with singular solutions instead of a

tutorial on solving such problems in general. Fermat had a copy of Arithmetica, and this is where he wrote his declaration that he had solved his

*Last Theorem* conjecture (see figure)

*Page 61 of Arithmetica by Diophantus. Fermat's "Last Theorem" note, as published in this edition, appeared after Problem VIII of Book II. Fermat's note reads in translation, "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." (Portion of a Wikimedia Commons image. Click for larger image.)*

Problem VIII of Book II in Arithmetica is interesting in its own right, since it explains how to

divide a square into a

sum of two squares. In the question, Diophantus takes the square to be 16. If the first of the summed squares is

**x**^{2}, then the second would be

**16 - x**^{2}. Diophantus then goes through a complicated method to find that

**x** = 16/5 (view

this Wikipedia page for details), with the result that one square is 256/25 and the other 144/25.

Diophantus also appears to have known that every number can be written as the sum of the squares of four integers, the proof of which is the 1770

Lagrange's four-square theorem by

Joseph-Louis Lagrange(1736-1813). This leads us to the conjecture involving a Diophantine equation that any number can be written as the sum of three cubes. The Diophantine equation is written

**k = x**^{3} + y^{3} + z^{3}, in which

**k** is the target number This idea seems strange at first until we remember that the cube of a negative number is also a negative number, so we can have very large numbers giving us small numbers in the summation.

A start in

computer-assisted search for solutions to the three cubes conjecture was a 1955 effort for

**0 ≤ k ≤ 100** and

**x, y, z** absolute magnitudes below 3164 by Miller and Woollett at the

Cambridge University Mathematical Laboratory using an

EDSAC computer in which they found 345 solutions.[1] A

multiplication on the EDSAC took 6

msec. A 2016 study by

Sander G. Huisman using about 100,000 hours of computer time on a variety of machines produced the first solution for

**z** = 74, which left just

**z** = 33 and

**z** = 43 as the last numbers up through 100 that were unsolved.[2] While 100 can be expressed quite simply with

**x** = -6,

**y** = -3. and

**z** = 7, the following table lists some of the larger number solutions for

**k** through 100. Reference 2 has a list of solutions for most values of

**k** up to 999.[2]

Sum of Three Cubes Conjecture
k |
x |
y |
z |

7 |
-168218632243945 |
-61605392145082 |
170928876351450 |

9 |
-579503130257826 |
47158155506281 |
579399014699414 |

20 |
-156759508867952 |
90524105730987 |
145971698874625 |

35 |
-244965503743381 |
82852212729479 |
241764631510633 |

46 |
-1042925392876187 |
-2525776940964 |
1042925397814257 |

55 |
-928487831742068 |
-272851005732953 |
936276531557204 |

63 |
-545743190147561 |
-287677064051260 |
571183926715614 |

70 |
-1381417714821543 |
-223954055560540 |
1383376964067653 |

74 |
-284650292555885 |
66229832190556 |
283450105697727 |

90 |
-455329473688079 |
225963644244842 |
435967889130081 |

97 |
-478725341478772 |
-438717423067543 |
579049714409178 |

99 |
-740582589889197 |
-123796510872653 |
741733872643349 |

100 |
-588182578216077 |
-183057438103967 |
594034558164966 |

A solution for

**k** = 33 was discovered in March, 2019, by

British mathematician.

Andrew R. Booker (b. 1976), who is

professor of

pure mathematics at the

University of Bristol (Bristol, UK). Booker found that 33 can be expressed as the sum of cubes of 8 866 128 975 287 528, −8 778 405 442 862 239, and −2 736 111 468 807 040, and that left 42 as the last

**k** through 100 that was unsolved.[4-5] Booker achieved this result by creating an

algorithm that searched about twenty times faster than

brute force.[4]

Science fiction fans will recognize that 42 is the

Answer to the Ultimate Question of Life, the Universe, and Everything, as found in the

The Hitchhiker's Guide to the Galaxy by

Douglas Adams (1952-2001).

The solution for

**k** = 42 was not long in coming, since a

collaboration between

Andrew Sutherland, a principal

research scientist in the

Department of Mathematics at MIT, and Andrew Booker discovered just a few months later that 42 can be expressed as the sum of cubes of -80 538 738 812 075 974, 80 435 758 145 817 515, and 12 602 123 297 335 631.[6] Their computer power was a worldwide set of more than 400,000

personal computers of

volunteers organized by the

UK-based Charity Engine.[6] There are now just ten numbers through 1000 that are unsolved, the next being 114.[6]

### References:

- J. C. P. Miller and M. F. C. Woollett, "Solutions of the Diophantine Equation: x3+y3+z3=k," J. London Mathematical Society, vol. 1-30, no. 1 (January, 1955), pp. 101-110, https://doi.org/10.1112/jlms/s1-30.1.101.
- Sander G. Huisman, "Newer sums of three cubes, arXiv, April, 26, 2016.
- Andrew R. Booker, "Cracking the problem with 33," arXiv, March 18, 2019.
- John Pavlus, "Sum-of-Three-Cubes Problem Solved for 'Stubborn' Number 33," Quanta Magazine, March 26, 2019.
- Numberphile, "42 is the new 33," YouTube Video, March 12, 2019
- Sandi Miller, "The answer to life, the universe, and everything," MIT Press Release, September 10, 2019.