
Griechische Mathematik: Geometrische Zahlen
Triangular numbers are so called because they can be arranged in a triangular array of dots (elements). Consider the sum 1+2+3+...+n = n(n+1)/2. This sum can be considered as the number of elements of a triangular number, see example with n = 3. For n =4 we have a special number for the Pythagoreans, the Tetraktys, that has 10 elements. I swear by the discoverer of the Tetraktys,Which is the spring of all our wisdom, The perennial root of Nature's fount. (Iambl., VP, 29.162)
1 + 3 = 2^{2 }, 1 + 3 + 5 = 3^{2}, 1 + 3 + 5 + 7 = 4^{2}, etc.
The image shows the first 3 triangular elements with 1,3 and 6 elements and their square are 1, 9 and 36 respectively. We have: 1^{2} = 1^{3}, 3^{2} = 1^{3} + 2^{3}, 6^{2} = 1^{3}+2^{3}+3^{3} , etc. This is equivalent to say that 1^{3}+2^{3}+...+ n^{3} = (n(n+1)/2)^{2} Square numbers are so called because they can be arranged as a square array of dots. The first four perfect numbers are: 6, 28, 496, 8128. Euclid was able to find that each of these numbers is of the form 2^{n}(2^{n+1} 1), where 2^{n+1}1 is prime. Euclid proved that all numbers of this form were perfect. The Pythagoreans knew that 1 + 2 + 4 + ... + 2^{k} = 2^{k+1}  1. Euclid's perfect numbers are triangular.
Divide the quadratic number in triangular numbers. We obtain (2n+1)^{2} = 8n(n+1)/2 + 1 Divide the quadratic number in two triangular numbers. n(n1)/2 + n(n+1)/2 = n^{2}. This is a graphic explanation why any pair of adjacent triangular numbers add to a square number.
Subtracting two quadratic numbers (n+1) and n we obtain, see figure above , (2n+1) = (n+1)^{2}  n^{2} Assume that there is a n for which 2n+1 = m^{2} (i.e. 2n+1 is a quadrat) and m is odd. Then n = (m^{2}1)/2, n+1 = (m^{2}+1)/2 or m^{2} + ( (m^{2}1)/2)^{2} = ((m^{2}1)/2)^{2}. We have therefore a method to produce integer solutions forming a Pythagorean triple. Every perfect number is triangular. The number 666, also known as the Number of the Beast, is a triangular number.
LINKS Triangular Numbers 1 and Triangular Numbers 2 Formulas and their geometric interpretation Polygonal numbers Patterns of prime distribution References There exist triangular numbers that are also square
What is a number? (From Rational , Irrational to Surreal Numbers)

