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Griechische Mathematik: Geometrische Zahlen
 Triangular number and sums
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)
 The sums of the uneven numbers, as reported by Aristotle in Metaphysics, give us the series of squared numbers:
1 + 3 = 2^{2 }, 1 + 3 + 5 = 3^{2}, 1 + 3 + 5 + 7 = 4^{2}, etc.
 Any pair of adjacent triangular numbers add to a square number
1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 45 + 55 = 100
The Pythagoreans were the first people to discover this relationship.
 According to Plutarch the square of the n^{th} triangular number equals the sum of the first n cubes.
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.
 Odd Square number relation to Triangular number
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.
 Pythagorean Numbers
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.
 Pentagonal Numbers
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

Eric W. Weisstein. "Figurate Number." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/FigurateNumber.html

Eric W. Weisstein. "Triangular Number." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/TriangularNumber.html
What is a number? (From Rational , Irrational to Surreal Numbers)
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