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As an example, the number 96 is a congruum because it is the difference between adjacent squares in the sequence 4, 100, and 196 (the squares of 2, 10, and 14 respectively). The first few congrua are:
All 14 squares in a 3×3-square (4×4-vertex) grid. As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common mathematical puzzle involves counting the squares in a large n by n square grid. [11] This count can be derived as follows: The number of 1 × 1 squares in the ...
For example, in this notation the sequence of even numbers could be written as (). The sequence of squares could be written as (). The variable n is called an index, and the set of values that it can take is called the index set.
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows:
Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 2 and can be written as 3 × 3.
The sequence can be used to prove that there are infinitely many prime numbers, as any prime can divide at most one number in the sequence. More strongly, no prime factor of a number in the sequence can be congruent to 5 modulo 6, and the sequence can be used to prove that there are infinitely many primes congruent to 7 modulo 12. [20]
Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to Pell's equation. The simplest example of this is the sequence of square triangular numbers.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.