Amongst the many joys of mathematics are the beautiful patterns that appear unexpectedly from time to time. In 1951, Alfred Moessner discovered a delightful and ingenious way of generating sequences of powers of natural numbers. It is surprisingly simple and offers great opportunities for development and generalization.
It is well-known that the sum of odd numbers yields a perfect square:
1 + 3 + 5 + … + (2n – 1) = n 2
This is easily demonstrated in a geometric way. We start with a unit square, and repeatedly add an additional row and column on the “east” and “north” sides and a unit square at the “north-east” corner. This amounts to adding the next odd number and, at each stage, a new square is produced.
Mossner’s Construction
We can generate the sequence of squares as the partial sums of the sequence of odd numbers. Moessner showed that a similar process can be used to generate the sum of kth powers for any k.
We start with the case k = 2. We write the sequence of natural numbers, 1, 2, 3, … and strike out every second number. Then we form the sequence of partial sums to obtain the sequence of perfect squares. This process is illustrated here.
Now for the case k = 3. Starting again with the natural numbers, we delete every third number and form the sequence of partial sums. From this new sequence, we delete every second number and again form partial sums. This gives the sequence of cubes.
Now for the case k = 4, starting again with the natural numbers, we delete every fourth number and form the sequence of partial sums. Now repeat, deleting every third entry, and then every second, to produce the sequence of fourth powers, as illustrated here:
The process can be generalized to produce a list of kth powers. We delete every k-th number and form partial sums. Then we delete every (k – 1)th number and again form partial sums. Repeating the process, we obtain the required sequence.
More Generalizations
The striking out and partial summing process is called Moessner’s construction, and it has been generalized in several ways. Instead of deleting evenly spaced numbers, we can vary the spaces. Suppose we delete the triangular numbers Tn = { 1, 3, 6, 10, 15, 21 } and form partial sums. Then we delete the final entry in each group and form sums again. Repeating this, we eventually produce the factorial numbers.
Other Possibilities
We do not have to start with the natural numbers. Long (1982) discusses many other possibilities. Students may derive much amusement from trying other combinations, varying the initial sequence and the pattern of deletion. As Long asks, “What happens if we delete the squares, the cubes or the Fibonacci numbers? What kinds of interesting results can be obtained with other special choices?”
A Remark on the Powers of the Natural Numbers. Moessner’s conjecture, published in 1951.
There is a reasonable prospect of discovering some new relationship. Even if it is already known, the excitement of personal discovery is undiminished. And even if you cannot prove a newly-found relationship, take comfort in the fact that Moessner never actually proved that his construction led to sequences of powers. The proof was first provided by Perron (1951). More general results were proved in Long’s paper.
Sources
[1] J. H. Conway and R. K. Guy, 1996: Moessner’s magic. In The Book of Numbers, pages 63–65. Springer-Verlag.
[2] C. T. Long, 1982: Strike it out–add it up. The Mathematical Gazette, 66(438):273–277.
[3] A. Moessner, 1951: Eine Bemerkung über die Potenzen der natürlichen Zahlen. Sitz. Bayer. Akad. Wiss., Math-Nat Kla. S.-B. 3, 29.
[4] O. Perron, 1951: Beweis des Moessnerschen Satzes. Sitz. Bayer. Akad. Wiss., Math-Nat Kla. 4:31–34.
ThatsMaths 