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
an + bn = cn
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unpublished proof around 1637.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
n2 = 4k2 + 4k + 1
n2 = 2(2k2 + 2k) + 1
so,
n2 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
x2, then the second would be
16 - x2. 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 = x3 + y3 + z3, 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.
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