A Cubed Positive Integer is a Sum of Consecutive Odd Numbers

by

Allen Klinger

 

Abstract                    Understanding and proving a proposition known to the early Greeks: The cube of every positive integer is a sum of consecutive odd integers.

 

Introduction          A positive integer n cubed always can be represented as the sum of consecutive odd numbers. Yet finding this decomposition can seem difficult, as is apparent from considering examples, e.g., "find 'cubes as odds' representations for 125 = 53, 343 = 73, or 117,649 = 3433." We give a descriptive method for this process and a formula with proof by mathematical induction.

 

Decompositions     Dividing numbers is a recurrent issue - see, e.g., [1, 2]. Perhaps that is why a frequent response to the question "what consecutive odd numbers sum to 73, i.e., 343?" is first puzzlement, and then sometimes a reply "how many odd numbers?" The response is not far off the mark due to the meaning of cube. Two examples from attempts at understanding the "cubes as odds" statement [3, 4] are:

 

 8  = 23 = 3 + 5                                                                                                (1)

27 = 33= 7 + 9 + 11                                                                                        (2)

 

Descriptions          With eight represented by two four-element square arrays, (1) is easily visualized. What then stands out that is that the expression 3 + 5 corresponds to removing one from the first four-element square array, and adding it to the second. A similar process for (2) leaves the middle of three nine-element square arrays unchanged, but removes two from the first, and adds them to the last.

The first procedure generalizes to any even positive integer cubed, while the second works for all cubed odd numbers. The even case involves taking a start position so half the square arrays comprising the cube lie on either side of a symmetry plane. Removal of one from the closest array, is followed by removing three from the next square array, five from the following one, and so on through all arrays on that side. Adding those removed numbers to arrays on the other side of that symmetry plane yields the numerical elements of the required sum. They are consecutive odd numbers by construction. For odd positive integers cubed, the symmetry plane is at the middle (as in the case of twenty-seven as three nine-element square arrays). Here the numbers removed from arrays on one side (added to those on the other) are two, then four, etc. Following this process quickly yields representations as in these examples:

 

216 = 63 = 31 + 33 + 35 + 37 + 39 + 41                                                         (3)

343 = 73 = 43 + 45 + 47 + 49 + 51 + 5 3+ 55                                                (4)

 

The process gives a constructive method that holds for any integer since any positive number whose cube we wish to represent is either even or odd. Clearly the construction yields n odd numbers that sum to the value of  n3; there are cases where fewer than n odds sum to  n3  so the decomposition is not unique. Issues of uniqueness, and visual or data structure [3, 5, 6] aspects, will be presented elsewhere.

 

Induction     Algebraically the statement "For any positive integer n,  n3 can be expressed as the sum of n consecutive odd integers," becomes:

n3  =  Si=1n [n (n - 1) + (2i - 1)]                                                             (5)

Expression (5) yields 1 = 13 = 1, and the statements at (1), (2), etc. The proof involves showing that truth of  (5) for n implies that it also holds for n + 1.

 

Proof          (n + 1)3                  =  n3  + 3n2 + 3n + 1

= Si=1n [n (n - 1) + (2i - 1)] + 3n2 + 3n + 1      [by (5)]

=  n[n(n - 1)] + [1 + 3 + ... + (2n -1)] + 3n2 + 3n + 1

=  [n3  + 2n2 + 3n + 1] + [1 + 3 + ... + (2n -1)]          

= [n3  + 2n2 + 3n + 1] - (2n+1) +  [1 + 3 + ... + (2n -1)] +(2n+1)

= (n2 + 2n +1)n + S i=1n+1 (2i - 1) = (n + 1)(n + 1)n + S i=1n+1 (2i - 1)

= Si=1n+1 [(n + 1)n + (2i - 1)]

Comment    "Cubes as odds" is from a first century A.D. source [4], quoted in [3]. Expressions (1), (2), and the first two paragraphs of Discussions, led to interaction with R. Blattner, P. Stelling, and others, one of whom, Bella Manel, provided the induction proof. P. Stelling raised issues about uniqueness and visual or data structure methods that we will present in a subsequent note.

 

[1] Gillings, Richard J., Mathematics in the Time of the Pharaohs, Cambridge, MA: The MIT Press, 1972; NY: Dover Publications, Inc., 1982, ISBN 0-486-24315-X.

 

[2] Stark, Harold M., An Introduction to Number Theory, The MIT Press, 1970, 1978, p. 5 ( Goldbach Conjecture).

 

[3] Naps, Thomas L., Introduction to Data Structures and Algorithm Analysis, 2nd Edition, West Publishing Co., St. Paul, Minnesota, 1992, p. 50, prob. 4.

 

[4 Nicomachus, Introduction Arithmetica, first century A.D. (see [3]).

 

[5] Knuth, Donald Erwin, The Art of Computer Programming: Fundamental Algorithms (Vol. I),  Addison-Wesley, Reading, Massachusetts, 1968.

 

[6] Klinger, Allen, "Data Structures", The Encyclopedia of Physical Science and Technology, Meyers, R. ed., NY: Academic Press, 1987, 125-135; revision Vol. 5, 43-56, 1992; revision, Vol. 4, 3rd Edition, 2001, pp. 263-276.