Oscar Wilde’s character Algernon said in The Importance of Being Earnest, “One must be serious about something, if one is to have any amusement in life.” Of course in Wilde’s typical ironic fashion, Algernon was only referring to his own dedication to frivolous diversions. In that spirit, allow me a few moments to tell a story about one of the odder sums of odd integers I discovered as a kid.
I remember that sometimes when I was bored — most especially during long, bi-weekly car trips with my parents — I would play various games with integers. I have no idea why, but at one point I memorized some huge list of powers of 2 (I can still remember the list from 1 to 65,536). I also computed the squares, cubes, and so forth of most of the smaller integers. As a result, I discovered on my own quite a number of interesting patterns in the integers. I don’t remember most of them, but there is one in particular that has stuck with me through the years.
First, I can’t see how someone can become a mathematician and not know the relation
I discovered that on my own during one of those car trips. It was very exciting at the time, though I will admit the magic has dulled somewhat in the last 15 years. It was around then when I devoted some time to thinking about sums of odd integers. During some of that time, I noticed something peculiar hidden in the above sum.
The sum of the next two odds is the second cube:
The sum of the next three odds is the third cube:
The sum of the next four odds is the fourth cube:
The sum of the next five odds is the fifth cube:
and so on.
That’s kind of surprising, right? Well, it’s less magical if you think about it. If this were true, we would immediately get the following equalities
The second equality follows from our well-known formula for perfect squares. That the third term equals the first follows by a trivial induction argument. This first equality, though, gives us exactly our magic pattern for cubes since . It is interesting to note that by the “Baby Gauss” formula, we have
It is easy to see that the pattern for general even powers of continues in a similar fashion since
The pattern for odds is similar in that you are adding strings of length . However, you start skipping some integers. For , we get
Skip 3. The sum of the next four odds is
Skip 13, 15, 17. The sum of the next nine odds is
Skip 37, 39, 41, 43, 45, 47. The sum of the next 16 odds is
By slightly changing our point of view, we describe the pattern more concretely. Suppose the odd power in question is . Consider the general sum of consecutive odd integers starting at given by
We can shift from the last term to the first, from the next-to-last term to the second, and so forth, to re-write the sum as
Since our sum should equal this gives immediately that
This formula agrees with what we have computed for . Observe that if we hit every integer for each , then we would get
which reduces to
and certainly only holds for .
In the even case, one can see that regrouping the sums provides an iteration scheme to get the sum of the first of the powers in terms of the lower order sums. However, by what we’ve just described, the situation gets more complicated for odd powers, and so such an endeavor is probably better abandoned. Indeed, Jacob Bernoulli used an algebraic method to deduce a smarter way of computing them and produced a closed form in terms of Bernoulli numbers. That seems a much more reasonable approach. I found a quote attributed to him where he discusses his opinion of what I am guessing is the regrouping method I just described:
“With the help of [these formulas] it took me less than half of a quarter of an hour to find that the 10th powers of the first 1000 numbers being added together will yield the sum
From this it will become clear how useless was the work of Ismael Bullialdus spent on the compilation of his voluminous Arithmetica Infinitorum in which he did nothing more than compute with immense labor the sums of the first six powers, which is only a part of what we have accomplished in the space of a single page.”
That’s got to sting a bit. And, mind you, Bernoulli and Bullialdus were contemporaries.