Sum Of Three Cubes
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In the mathematics of
sums of powers In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's thre ...
, it is an
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...
to characterize the numbers that can be expressed as a sum of three
cubes In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
of integers, allowing both positive and negative cubes in the sum. A necessary condition for n to equal such a sum is that n cannot equal 4 or 5
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...
9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient. Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non-zero
natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...
.


Small cases

A nontrivial representation of 0 as a sum of three cubes would give a
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
to
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefore, by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
's proof of that case of Fermat's last theorem, there are only the trivial solutions :a^3 + (-a)^3 + 0^3 = 0. For representations of 1 and 2, there are infinite families of solutions :(9b^4)^3+(3b-9b^4)^3+(1-9b^3)^3=1 (discovered by K. Mahler in 1936) and :(1+6c^3)^3+(1-6c^3)^3+(-6c^2)^3=2 (discovered by A.S. Verebrusov in 1908, quoted by L.J. Mordell). These can be scaled to obtain representations for any cube or any number that is twice a cube. For 1, there exist other representations and other parameterized families of representations. For 2, the other known representations are :1\ 214\ 928^3 + 3\ 480\ 205^3 + (-3\ 528\ 875)^3 = 2, :37\ 404\ 275\ 617^3 + (-25\ 282\ 289\ 375)^3 + (-33\ 071\ 554\ 596)^3 = 2, :3\ 737\ 830\ 626\ 090^3 + 1\ 490\ 220\ 318\ 001^3 + (-3\ 815\ 176\ 160\ 999)^3 = 2. However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials as above. Even in the case of representations of 3, Louis J. Mordell wrote in 1953 "I do not know anything" more than its small solutions :1^3+1^3+1^3=4^3+4^3+(-5)^3=3, other than the fact that in this case each of the three cubed numbers must be equal modulo 9.


Computational results

Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations. used a method of involving
lattice reduction In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least exponent ...
to search for all solutions to the
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
:x^3+y^3+z^3=n for positive n at most 1000 and for \max(, x, ,, y, ,, z, )<10^, leaving only 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975 as open problems in 2009 for n\le 1000, and 192, 375, and 600 remain with no primitive solutions (i.e. \gcd(x,y,z)=1). After
Timothy Browning Timothy Browning is a mathematician working in number theory, examining the interface of analytic number theory and Diophantine geometry. Browning is currently a Professor of number theory at the Institute of Science and Technology Austria (ISTA ...
covered the problem on
Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
in 2016, extended these searches to \max(, x, ,, y, ,, z, )<10^ solving the case of 74, with solution :74=(-284\ 650\ 292\ 555\ 885)^3+66\ 229\ 832\ 190\ 556^3+283\ 450\ 105\ 697\ 727^3. Through these searches, it was discovered that all n < 100 that are unequal to 4 or 5 modulo 9 have a solution, with at most two exceptions, 33 and 42. However, in 2019, Andrew Booker settled the case n=33 by discovering that :33=8\ 866\ 128\ 975\ 287\ 528^3+(-8\ 778\ 405\ 442\ 862\ 239)^3+(-2\ 736\ 111\ 468\ 807\ 040)^3. In order to achieve this, Booker exploited an alternative search strategy with running time proportional to \min(, x, ,, y, ,, z, ) rather than to their maximum, an approach originally suggested by Heath-Brown et al. He also found that :795=(-14\ 219\ 049\ 725\ 358\ 227)^3 + 14\ 197\ 965\ 759\ 741\ 571^3 + 2\ 337\ 348\ 783\ 323\ 923^3, and established that there are no solutions for n=42 or any of the other unresolved n \le 1000 with , z, \le 10^. Shortly thereafter, in September 2019, Booker and Andrew Sutherland finally settled the n=42 case, using 1.3 million hours of computing on the
Charity Engine Charity Engine is a free PC app based on Berkeley University's BOINC software, run by The Worldwide Computer Company Limited. The project works by selling spare home computing power to universities and corporations, then sharing the profits betwe ...
global grid to discover that :42=(-80\ 538\ 738\ 812\ 075\ 974)^3 + 80\ 435\ 758\ 145\ 817\ 515^3 + 12\ 602\ 123\ 297\ 335\ 631^3, as well as solutions for several other previously unknown cases including n=165 and 579 for n \le 1000. Booker and Sutherland also found a third representation of 3 using a further 4 million compute-hours on Charity Engine: :3 = 569\ 936\ 821\ 221\ 962\ 380\ 720^3 + (-569\ 936\ 821\ 113\ 563\ 493\ 509)^3 + (-472\ 715\ 493\ 453\ 327\ 032)^3. This discovery settled a 65-year old question of Louis J. Mordell that has stimulated much of the research on this problem. While presenting the third representation of 3 during his appearance in a video on the Youtube channel
Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
, Booker also presented a representation for 906: :906 = (-74\ 924\ 259\ 395\ 610\ 397)^3 + 72\ 054\ 821\ 089\ 679\ 353\ 378^3 + 35\ 961\ 979\ 615\ 356\ 503^3. The only remaining unsolved cases up to 1,000 are the seven numbers 114, 390, 627, 633, 732, 921, and 975, and there are no known primitive solutions (i.e. \gcd(x,y,z)=1) for 192, 375, and 600.


Popular interest

The sums of three cubes problem has been popularized in recent years by
Brady Haran Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Periodic Videos'' and ''Number ...
, creator of the
YouTube YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by ...
channel
Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
, beginning with the 2015 video "The Uncracked Problem with 33" featuring an interview with
Timothy Browning Timothy Browning is a mathematician working in number theory, examining the interface of analytic number theory and Diophantine geometry. Browning is currently a Professor of number theory at the Institute of Science and Technology Austria (ISTA ...
. This was followed six months later by the video "74 is Cracked" with Browning, discussing Huisman's 2016 discovery of a solution for 74. In 2019, Numberphile published three related videos, "42 is the new 33", "The mystery of 42 is solved", and "3 as the sum of 3 cubes", to commemorate the discovery of solutions for 33, 42, and the new solution for 3. Booker's solution for 33 was featured in articles appearing in ''
Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. ''Undark Magazine'' described ''Quanta Magazine'' as "highly regarded for ...
'' and ''
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'', as well as an article in ''
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'' in which Booker's collaboration with Sutherland was announced: "...the mathematician is now working with Andrew Sutherland of MIT in an attempt to find the solution for the final unsolved number below a hundred: 42". The number 42 has additional popular interest due to its appearance in the 1979
Douglas Adams Douglas Noel Adams (11 March 1952 – 11 May 2001) was an English author and screenwriter, best known for ''The Hitchhiker's Guide to the Galaxy''. Originally a 1978 BBC radio comedy, ''The Hitchhiker's Guide to the Galaxy'' developed into a " ...
science fiction novel ''
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'' as the answer to The Ultimate Question of Life, the Universe, and Everything. Booker and Sutherland's announcements of a solution for 42 received international press coverage, including articles in ''New Scientist'', ''
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'', ''
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'', ''
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'', ''
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'', ''
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'', ''
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'', ''
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,
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, and
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. ''Popular Mechanics'' named the solution for 42 as one of the "10 Biggest Math Breakthroughs of 2019". The resolution of Mordell's question by Booker and Sutherland a few weeks later sparked another round of news coverage. In Booker's invited talk at the fourteenth
Algorithmic Number Theory Symposium Algorithmic Number Theory Symposium (ANTS) is a biennial academic conference, first held in Cornell in 1994, constituting an international forum for the presentation of new research in computational number theory. They are devoted to algorithmic a ...
he discusses some of the popular interest in this problem and the public reaction to the announcement of solutions for 33 and 42.


Solvability and decidability

In 1992,
Roger Heath-Brown David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervis ...
conjectured that every n unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes. The case n=33 of this problem was used by
Bjorn Poonen Bjorn Mikhail Poonen (born 27 July 1968 in Boston, Massachusetts) is a mathematician, four-time Putnam Competition winner, and a Distinguished Professor in Science in the Department of Mathematics at the Massachusetts Institute of Technology. Hi ...
as the opening example in a survey on
undecidable problem In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an ...
s in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, of which
Hilbert's tenth problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equat ...
is the most famous example. Although this particular case has since been resolved, it is unknown whether representing numbers as sums of cubes is decidable. That is, it is not known whether an algorithm can, for every input, test in finite time whether a given number has such a representation. If Heath-Brown's conjecture is true, the problem is decidable. In this case, an algorithm could correctly solve the problem by computing n modulo 9, returning false when this is 4 or 5, and otherwise returning true. Heath-Brown's research also includes more precise conjectures on how far an algorithm would have to search to find an explicit representation rather than merely determining whether one exists.


Variations

A variant of this problem related to
Waring's problem In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural num ...
asks for representations as sums of three cubes of non-negative integers. In the 19th century,
Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
and collaborators compiled tables of solutions to this problem. It is conjectured that the representable numbers have positive
natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...
. This remains unknown, but
Trevor Wooley Trevor Dion Wooley FRS (born 17 September 1964) is a British mathematician and currently Professor of Mathematics at Purdue University. His fields of interest include analytic number theory, Diophantine equations and Diophantine problems, h ...
has shown that \Omega(n^) of the numbers from 1 to n have such representations. The density is at most \Gamma(4/3)^3/6\approx 0.119. Every integer can be represented as a sum of three cubes of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s (rather than as a sum of cubes of integers).


See also

*
Sum of four cubes problem The sum of four cubes problem asks whether every integer is the Summation, sum of four Cube (algebra), cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proven nor disproven. Some of the cubes m ...
, whether every integer is a sum of four cubes * , relating to cubes that can be written as a sum of three positive cubes *
Plato's number Plato's number is a number enigmatically referred to by Plato in his dialogue the ''Republic'' (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no rea ...
, an ancient text possibly discussing the equation 3 + 4 + 5 = 6


References


External links


Solutions of for {{math, 1=0 ≤ ''n'' ≤ 99
Hisanori Mishima

Daniel J. Bernstein
Sums of three cubes
Mathpages Additive number theory Diophantine equations Unsolved problems in number theory