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Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone, [2] and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. [3]
The two right triangles with leg and hypotenuse (7,13) and (13,17) have equal third sides of length .The square of this side, 120, is a congruum: it is the difference between consecutive values in the arithmetic progression of squares 7 2, 13 2, 17 2.
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as ...
Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares) The British flag theorem for rectangles equates two sums of two squares; The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals
The sequence of squares could be ... is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define ...
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.
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 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]