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In the mathematics of
Lucas conjectured that the only solutions are ''N'' = 1, ''M'' = 1, and ''N'' = 24, ''M'' = 70, using either 1 or 4900 cannon balls. It was not until 1918 that G. N. Watson found a proof for this fact, using
In the mathematics of 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 term can mean
* polygon ...
s, the cannonball problem asks which numbers are both square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
and square pyramidal
In molecular geometry, square pyramidal geometry describes the shape of certain Chemical compound, compounds with the formula where L is a ligand. If the ligand atoms were connected, the resulting shape would be that of a Square pyramid, pyram ...
. 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 cont ...
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 rebellion ...
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
Lucas ...
formulated the cannonball problem as a 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 ...
:
or
:
Solution
Lucas conjectured that the only solutions are ''N'' = 1, ''M'' = 1, and ''N'' = 24, ''M'' = 70, using either 1 or 4900 cannon balls. 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 a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
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 ...
. The result has relevance to the bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the c ...
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 Cannon Ball Problem, which is to yield a perfect square from the ''N''th Tetrahedral number, 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 cannon balls. Similarly, a pentagonal-pyramid version of the Cannon Ball problem to produce a perfect square, would have ''N'' = 8, yielding a total of (14 × 14 = ) 196 cannon balls. The only numbers that are simultaneouslytriangular
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- collinea ...
and square pyramidal, are 1, 55, 91, and 208335.
There are no numbers (other than the trivial solution 1) that are both tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
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 perfect square. There are infinitely many square triangular numbers; the first few are:
:0, 1, 36, , , , , , ,
Expl ...
, the numbers that are simultaneously square and triangular
*Sixth power In arithmetic and algebra the sixth power of a number ''n'' is the result of multiplying six instances of ''n'' together. So:
:.
Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourt ...
, the numbers that are simultaneously square and cubical
*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). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
References
External links
*{{mathworld, id=CannonballProblem, title=Cannonball Problem Diophantine equations Figurate numbers