Number

2021-01-29   |   by CusiGO

Little Gauss’s mental trick to add the first 100 numbers is, as we saw last week, to make 50 pairs of them, which add up to the first and last pair: (100 + 1) x 50 = 5050. When n numbers are generalized, their sum s will be: S = (n + 1) n / 2, which is the formula to give the numerical value of the nth triangle. Therefore, given the number of any two continuous triangles, that is, the nth and Nth 1, the sum of them should be:

(N+1)N/2+(N+2)(N+1)/2=N²/2+N/2+N²/2+N+N/2+1=N²+2N+1=(N+1)²

Then the sum of two consecutive triangles is a perfect square, that is, the square of the larger of the two numbers. It’s easy to display graphically, because two consecutive triangle numbers can be “coupled” into a square, as shown in the figure, in which the third triangle number and the fourth triangle number constitute the fourth square number.

A similar graphical representation helps us to see that N 2 is the sum of N first odd numbers (nicomache’s theorem).

There are different ways to prove that every perfect square has an odd divisor. Our usual commentator, Salva foster, provides the following:

To see that N 2 has odd divisors, just note that on the one hand, n is a divisor, because NxN = N 2; on the other hand, if it has more divisors, they always appear in pairs, because if a x B = N 2, a and B are different from n, one of them is greater than N, and the other is smaller.

Therefore, if a and B are different, then N2 will have one or more pairs of divisors different from n plus N itself, that is, odd divisors; if we add N 2 itself and 1 to the divisor, their number will still be odd.

Triangle and square numbers are the most famous members of the family of figure numbers. In view of the synonyms of “image” and “imagination”, people may think that they are related to imaginary numbers, but the fact is just the opposite: imaginary numbers are unthinkable, and imaginary numbers are called “images” because they are easily regarded as geometric figures. Because they are boundary elements between arithmetic and geometry, some ancient great mathematicians, such as Pythagoras, diophant of Alexandria and nicomachos of gerassa, studied them.

Among figure numbers, polygon numbers are particularly interesting. Polygon numbers can be expressed as a set of equidistant points that constitute regular polygons. The first polygon numbers are triangles and squares that we’ve talked about in recent weeks. According to the same standard, we can form pentagons, hexagons, heptagoniums…

As we can see, the sequence of triangles is 1, 3, 6, 10, 15, 21… The sequence of squares is 1, 4, 9, 16, 25, 36… What is the sequence of pentagons? How about hexagon? Is there a significant relationship between continuous polygon number sequences?

Carlo frebetty is a writer and mathematician at the New York Academy of Sciences. He has published more than 50 popular science books for adults, children and teenagers, including damned physics, damned mathematics or great games. He’s a screenwriter for crystal ball.

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