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Griechische Mathematik: Geometrische Zahlen

Triangular number and sums

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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 = 2², 1 + 3 + 5 = 3², 1 + 3 + 5 + 7 = 4², 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.

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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:

1² = 1³, 3² = 1³ + 2³, 6² = 1³+2³+3³ , etc.

This is equivalent to say that 1³+2³+...+ n³ = (n(n+1)/2)²

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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 2ⁿ(2ⁿ⁺¹- 1), where 2ⁿ⁺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

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Divide the quadratic number in triangular numbers. We obtain (2n+1)² = 8n(n+1)/2 + 1

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Divide the quadratic number in two triangular numbers. n(n-1)/2 + n(n+1)/2 = n². This is a graphic explanation why any pair of adjacent triangular numbers add to a square number.