In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square 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.
When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America. [1] Édouard Lucas formulated the cannonball problem as a Diophantine equation
or
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 functions. More recently, elementary proofs have been published. [2] [3]
The solution N = 24, M = 70 can be used for constructing the Leech lattice. The result has relevance to the bosonic string theory in 26 dimensions. [4]
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.
A triangular-pyramid version of the Cannon Ball Problem, which is to yield a perfect square from the Nth 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. [5]
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. [6]
The only numbers that are simultaneously triangular and square pyramidal, are 1, 55, 91, and 208335. [7] [8]
There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal. [8]
François Édouard Anatole Lucas 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.
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
In mathematics, a square triangular number is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
300 is the natural number following 299 and preceding 301.
360 is the natural number following 359 and preceding 361.
2000 is a natural number following 1999 and preceding 2001.
10,000 is the natural number following 9,999 and preceding 10,001.
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is,
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.
3000 is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English.
4000 is the natural number following 3999 and preceding 4001. It is a decagonal number.
5000 is the natural number following 4999 and preceding 5001. Five thousand is the largest isogrammic numeral in the English language.
7000 is the natural number following 6999 and preceding 7001.
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The th octahedral number can be obtained by the formula:
A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with i2 points on the square faces of the ith layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has n + 1 points along each of its edges.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form n(2n2 − 1).
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So:
