Monday, 17 December 2012

Sums of Powers of Integer Sequences

A short while back I mentioned David Well's new book, Games and Mathematics, and talked about the beautiful theorem of Liouville about the sets of numbers for which the sum of the numbers cubed is equal to the square of their sum.
One of the sets is the set of positive integers. (13+23+33+... +n3)= (1+2+3+...+n)2.

This is one of three identities (the sum of the integers, the sum of the squares of the integers, and the sum of the cubes of the integers) that show up in pre-calc classes that students are often challenged to memorize, and prove by induction.

I was surprised recently to learn that the simple method I always showed my students was NOT the way the ancients first knew this expression.

I had always assumed that due to their incredible interest in figurate numbers, the early Greeks/Romans/Arabs knew that if any two consecutive triangular numbers are squared, their difference is a cube. 32 - 12 = 23, and if we jump up to the 5th triangular number 15, and the 4th, 10, and take the difference of their squares, the result is 152 - 102 = 53 or 125. It's even pretty easy to show that the result is true wtih algebra using the (well known to the Greeks) fact that the Nth triangular number is 1/2 (n)(n+1).

If we use the notation T(n) to represent the nth triangular number then we can write 13+23+33 +... + n3 as
T2(1) + T2(2)-T2(1) + .... + T2(n)-T2(n-1) and it is clear that every term except the last cancels, giving us the sumo of the first n cubes is the square of the nth triangular number.

But it seems that Nichomachus, who lived around 100 AD, presented the sums of cubes by looking at the sequence of odd numbers which he called the gnomens of squares. (A gnomen is the greek name for a device similar to today's common carpenter's square. If you take a square array of points, such as 3 2shown, adding the fourth odd number, 7, produces the next square. This arrangement of points or squares was called a gnomen.

As early as the Pythagoreans it was well known that the sequence of n odd numbers produced a square number. 1+3+5+7 =42
To discuss the sum of cubes, Nichomachus points out that if you start with 1,3,5,7,9,11,13... you note that the first (1) is a cube, and the next two (3+5=8) are a cube, and the next three, 7+9+11 = 27 are a cube. As each cube is added, we add additional gnomens to maintain a square number.
Since the number of gnomens added was a triangular number, 1 + 2 + 3 etc odd gnomens, they must add up to the square of a triangular number, and as pointed out above, they had known for half a millennium that the first n triangular numbers added up to 1/2 (n)(n+1) and so the square of that number, would be the same as the sum of the first n cubes.

I wanted to point out a little about the history of the sum of the squares of the integers, which seems to include a very non-intuitive 2n+1 term that seems to be out of place, and a general way I use to derive that formula that students seen not to see, but I will save that for a day or two away.




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