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In the mathematics of
Lucas conjectured that the only solutions are , , and , using either 0, 1, or 4900 cannonballs. It was not until 1918 that G. N. Watson found a proof for this fact, using
figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...
s, the cannonball problem asks which numbers are both square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid? Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1?
Formulation as a Diophantine equation
When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number;Thomas Harriot
Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his con ...
gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh
Sir Walter Raleigh (; – 29 October 1618) was an English statesman, soldier, writer and explorer. One of the most notable figures of the Elizabethan era, he played a leading part in English colonisation of North America, suppressed rebell ...
on their expedition to America.
Édouard Lucas
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François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
Biography
Luc ...
formulated the cannonball problem as a Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
:
or
:
Solution
Lucas conjectured that the only solutions are , , and , using either 0, 1, or 4900 cannonballs. It was not until 1918 that G. N. Watson found a proof for this fact, using elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
s. More recently, elementary proof In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain ...
s have been published.
Applications
The solution ''N'' = 24, ''M'' = 70 can be used for constructing theLeech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Er ...
. The result has relevance to the bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the context of string theory, and a new ve ...
in 26 dimensions.
Although it is possible to tile a geometric square with unequal squares, it is not possible to do so with a solution to the cannonball problem. The squares with side lengths from 1 to 24 have areas equal to the square with side length 70, but they cannot be arranged to tile it.
Related problems
A triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the ''N''thtetrahedral number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid (geometry), pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular ...
, would have ''N'' = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (702 × 22 = 1402 = ) 19600. This is comparable with the 24th square pyramid having a total of 702 cannonballs. This is because a square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part ...
is one-fourth of a larger tetrahedral number (meaning the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron with twice as many points along each edge).
Similarly, a pentagonal-pyramid version of the cannonball problem to produce a perfect square, would have ''N'' = 8, yielding a total of (14 × 14 = ) 196 cannonballs.
The only numbers that are simultaneously triangular
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...
and square pyramidal are 1, 55, 91, and 208335.
There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.
See also
*Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the sum of all integers from 1 to n has a square root that is an integer. There are inf ...
, the numbers that are simultaneously square and triangular
*Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or Lattice (group), lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fract ...
References
External links
*{{mathworld, id=CannonballProblem, title=Cannonball Problem Diophantine equations Figurate numbers