Number Patterns and Mathematical Knowledge
Keywords Numbers, Sums, Square Pyramid, Exponents
Abstract Sum/sum-of-squares identities. Using a simple algebraic fact. Consecutiveinteger relationships. Patterns and symmetry. Computer exploration and math
learning.
1
Introduction
This article concerns examples that are possible starting points for a school lesson.
The idea is to present them in connection with related mathematics topics. In
discovering the general truths implied by the examples, lessons can be drawn to
highlight fundamental notions - proof, identities, and induction - or to develop
skill with algebra.
Start with the Pythagorean theorem and Fermat’s conjecture, both below in
equation form:
x2 + y 2 = z 2
(1)
xn + y n = z n
(2)
There is a well-known number-relation that anchors this paper. It is only one
of many numerical cases that satisfy equation (1), but it plays a dual role later, so
we present it here:
x = 3, y = 4, z = 5
(3)
Many other interesting number-relations exist; two are:
102 + 112 + 122 = 132 + 142
(4)
21 + 23 + 25 + 27 + 29 = 53
(5)
Someone curious about (1) and (2) might ask a question like these. How many
terms should appear in an expression involving squares summed? What do we
know about relationships of numbers raised to the same or different powers and
then added?
1
2
Discovery
With rope, a straight stick, and knowledge [basically (3), (1) and its geometric
meaning] it is possible to build a pyramid with 90◦ corners: see [18]. The numerical
patterns in the many triples satisfying (1) are fascinating. Internet exposition
about them [5] is complete and accessible. That means that teachers can use both
examples and on-line text to offer new modes, interesting ways for people to go
beyond names like Pythagorean triple and explore on their own in learning math.
Much different series of numerical examples follow from the following two statements. The first is from a magazine column puzzle [2], while the second is from
ancient knowledge transmitted via a textbook’s preliminary statement in an exercise [12]:
We are interested in finding a sequence of 2n + 1 consecutive positive
integers, such that the sum of the squares of the first n + 1 integers
equals the sum of the squares of the last n integers. The simplest such
sequence is
32 + 42 = 52
(6)
The second goes beyond squares, raising the issue of higher powers:
The cube of every positive integer is a sum of consecutive odd integers.
[13]
The sources for [2] are [11], an unavailable letter [16] mentioned there, and
[17]. The last presents the first four equations involving summing sequences of
adjacent integers-squared, while [11] presents the first five such by making use of
an observation from [16]. All these appear in Table 1.
Table 2 shows how four specific cubed positive integers, 5, 6, 7, and 8 can be
written as sums of consecutive odd integers. (To make the mystery begin to vanish
view [9].)
Pattern thinking removes the magic from the expressions in the above two
tables. The way to understand what is happening in these two situations precedes
using words like sequences of adjacent integers-squared. We will see below that
physical instances initially led to these and other observations.
If these examples are put forward as discussion topics, students’ observations,
their awareness of evident patterns in these tables, can lead back to topics used
in the mathematics curriculum, specifically, mathematical induction and algebraic
reasoning.
2
32 + 42 = 52
(7)
102 + 112 + 122 = 132 + 142 = 365
(8)
212 + 222 + 232 + 242 = 252 + 262 + 272
(9)
362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442
(10)
552 + 562 + 572 + 582 + 592 + 602 = 612 + 622 + 632 + 642 + 652
(11)
Table 1: Consecutive Squared-Integer Sums
3
21 + 23 + 25 + 27 + 29 = 125 = 53
(12)
31 + 33 + 35 + 37 + 39 + 41 = 216 = 63
(13)
43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 = 73
(14)
57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512 = 83
(15)
Table 2: Cubes as Sums of Consecutive Odd Numbers
3
Observations
The first three Table 1 equations [(7) - (9)] motivate the following question. What
value begins the series of consecutive numbers that are squared? (Gardner wrote
that Linton [16] gave a formula for the lowest value in such an equation. But
Gardner’s book doesn’t contain it and Linton’s letter to him isn’t available.)
Instead of the lowest value, view this through pattern thinking. First look at
the these equations in terms of symmetry. In each, there is one more on the left
side than there are terms right of the equals sign. The symmetry is about the
largest of the numbers (before squaring) on the left hand side. In essence the
symmetry comes from counting down from that number on the left, and up from
it on the right.
Suppose one divides the numbers that are squared into three groups: those
lower than the value closest to, and left of, the equals sign; that number itself; and
those on the right side of the equality.
In (7) - (9), but actually any consecutive squares equation, all terms are derived
by counting from the particular value we just isolated (one of three groups). That
value is always immediately left of the equals sign when there is one more term at
its left than on the right. If we call the isolated value (with location just left of
the equals sign when the larger number of consecutive squares is on the left side)
the pivot, numbers to be squared on the left are obtained by counting down from
4
it (those at right, up).
There are other expressions not involving squares that are very similar to the
above equations, for example (16) - (18) . In all these cases there is a single number
from which all others in that equation are found by counting down or counting up.
Equation (19) states algebraically what takes place in the equations (16) - (18)
, using a to represent the largest offset from the pivot term t.
1+2=3
(16)
4+5+6=7+8
(17)
9 + 10 + 11 + 12 = 13 + 14 + 15
(18)
(t − a) + (t − a + 1) + ... + t = (t + 1) + ... + (t + a)
(19)
Every term on the RHS of (19) has (after the letter t) some value between one
and a; hence there are a-many entries there. Similar inspection of the terms on
the LHS of (19) shows that there are a + 1 terms left of the equals sign.
Expanding the elements in(19) lets us collect all terms with t’ s on the LHS,
and all without t’ s on the RHS. That leads to these two equalities [equation (20).
(a + 1)t − a(t) = t = 2(sum − of − numbers − f rom − 1 − to − a)
(20)
Rewriting sum-of-numbers-from-1-to-a using summation notation, and division
through by 2 yields this restatement of equation (20)
t/2 =
a
X
i = a(a + 1)/2
(21)
i=1
To check, substitute a = 3 in equation (21) and find the equation (18) pivot value.
Equation (21) is often demonstrated by writing two rows of numbers 1 through
a, one below the other but in reverse order. Vertically adding similarly-located
values gets just a entries that all are (a + 1) in value, hence [since that addition
yields sum-of-the-numbers-from-1-to-a twice] :
2
a
X
i = a(a + 1)
(22)
i=1
To summarize, in an expression like (16)-(18) a number just left of the equals
sign (side with larger number of terms) has value t . The value t begins a counting
5
down and up process that forms the equality. Finally, t can be calculated from
the number of right (fewer entries) side terms, a , by:
t = a(a + 1)
(23)
We count down a-many times to get the base which we will call b. Hence for
linear [exponent one, general form (19)] cases, b is reached from subtracting a from
t:
b = t − a = a(a + 1) − a = a2
4
(24)
Squares
The linear/exponent-one case explained above makes the [2] question about consecutive squares like (7) - (9) straightforward. While we will get to the lowest
value b formula, the best approach begins by finding the pivot value. The reasoning begins with an expression analogous to (19) stating the consecutive square
property:
(t − a)2 + (t − a + 1)2 + ... + t2 = (t + 1)2 + ... + (t + a)2
(25)
Again three groups of numbers appear. One group is just the pivot, t . The
other two, are numbers below it, and those that are higher. The values to be
squared in these last two groups come from differences or sums to t (they count
down or up from it). That fact leads to using:
(c − d)2 = c2 − 2cd + d2 = (c2 + 2cd + d2 ) − 4cd = (c + d)2 − 4cd
(26)
Applying (26) to (25) leads to:
2
t =4
a
X
ti
(27)
i
(28)
i=1
which simplifies:
t=4
a
X
i=1
From the expression for sums of integers from 1 to n, (21), the result in (29)
can be restated. In the linear case equation (23) results. Here applying (22) to
(28) gives twice that:
t=4
a
X
i = 2a(a + 1)
i=1
6
(29)
Just as in the linear case, we can use the observation that we count down from
the pivot a, the number of terms appearing on the right side, to find the base b.
That implies Linton’s formula was:
b = t − a = 2a(a + 1) − a = 2a2 + a = a(2a + 1)
(30)
Direct substitution of a = 1, 2, 3 in equation (30) yields b = 3, 10, 21.
This and the previous section used a form of symmetry analysis and algebraic
properties (expanding the expressions from squaring a) the sum, and b) the difference, of two terms).
5
Values
This section illustrates number-relationships involving equations of form (25). The
general notion is that people see things about sequences of adjacent numbers
squared and summed. In a teaching environment these results and interconnections
can be used to stimulate students by drawing on their observations and pattern
awareness skill. The key patterns involve symmetry, order - a slight generalization
of what is above described by a, number of terms on right side; and total number
of entries, i.e., (2a + 1).
Instead of a let order signify smallest number of terms in a relationship between
two sets of summed squares on consecutive values (this eliminates the “right of
the equals sign” restriction). A reasonable word for the symbol b is either base
or bottom: this refers to each consecutive-square equation’s lowest-value-beingsquared.
The observed values for order and base of equations (7) - (9) follow in Table 2.
Notice that the product of order and number of entries is the base (entries refers
to the total number of terms or elements in an equation). Stated algebraically,
this is Linton’s formula, (30), namely a(2a + 1) = b.
order
1
2
3
base
3
10
21
number of entries
3
5
7
Table 3: Product Pattern
Table 4 shows symmetry about the pivot. It also displays pattern relationships
between characteristics of different order equations.
Notice the result from multiplying two items and dividing by a third. Those
to be multiplied are number of left side terms and largest value on right. Division
7
(←
count
21
10
22
down)
3
11
23
pivot
4
12
24
= (count
=
5
=
13
=
25
up →)
14
26
27
Table 4: Symmetry Pattern
is by number of right side terms. The result is exactly base of next-higher-order
equation.
Also, notice that right side counting up from a pivot makes the largest value be
t + a. The result from multiplying two properties, dividing by a third, can be put
in algebraic terms. We do that using subscripts that let us know which equation
we are at, either a or a + 1 order:
(ta + aa )(aa + 1)/aa = ba+1
(31)
Since (30) holds in general, (31) can be rewritten using ba+1 = (a + 1)(2a + 3) :
(t + a)(a + 1)/a = (a + 1)(2a + 3)
(32)
Algebraic manipulation here again leads to t = 2a(a + 1) as in (29).
6
Cubes as Sum of Adjacent Odds
Up to now this paper has put forth connections between linear and squared terms.
It early on mentioned a past issue: representing a cube, as in the Nicomachus
statement [12].
Again this is a situation where numeric examples are easily brought forth, one
where hand calculators may be employed, and a candidate for instruction about
proof. You can establish that the statement holds for all positive integers using
mathematical induction [3], [4]. But to do that you need to change by leaving the
symbolic realm.
This topic requires visual thinking. It is difficult to understand within the
domain of mathematical symbols for exponents. But cube has real physical meaning. You can construct a model using eight or twenty-seven dice to explain 23
and 33 . Another method is to visualize either of two things. A Cartesian system with three coordinates handles 23 by providing the eight octants. Likewise a
three-dimensional tic-tac-toe game represents 33 .
The last paragraph, math-induction, and recognizing the difference between
numbers being cubed that are odd and even could lead to the ability to write
down expressions like (12)-(15) for any positive integer cubed. Your students can
8
learn to do that and then go out into the world armed with the puzzling (to others)
question, how can you represent ... as the sum of adjacent odd numbers? Whether
it leads to it rapidly or not depends on the prior paragraph, and access to such
material as [1], [15], [8] or reference to [10], [9] .
7
Square Sum of Summed Squares
This issue involves piling spheres as done with marbles in the photo. The pile
creates a pyramid with square cross -section: one at the top, a two-by-two array
below that, and so forth to the n-th level (n2 ). The goal is to choose n so that for
some integer m the next equation holds:
12 + 22 + ... + n2 = m2
(33)
The word description of the preceding equation is that a series of consecutive
squares summed begins with unity and the result is a square. As the number of
elements summed increases, an interesting fact appears. The total is only square
when the last number squared is twenty-four [7]. (For more detail on this question
see [6]; proof is beyond the scope of this item.) In mathematical statements:
1) there is an integer-pair (n, m) satisfying (33) (existence); and,
2) there is only one integer-pair (n, m) satisfying (33) (uniqueness).
These statements show two ways mathematicians look at situations.
Another issue is behind this. The expression (21) or its alternate form (22) and
possibly (26) are thought of as truths designated as important by being labeled
identities. There is another identity, a summing squares relationship that equates
the sum of squares beginning at one and on up through the integers to a value n,
as the product of three factors in n:
12 + 22 + ... + n2 = n(n + 1)(2n + 1)/6
(34)
The values 12, 18, 24, 30 and 36 all are evenly divided by 6. Of them, the
only one that could be an n yielding equation (34) square is n = 24. The following
table shows this.
Although we haven’t established the uniqueness of n = 24 as the only value
solving 34 that has been done elsewhere; see [6]. Whether one seeks materials
for communicating further using numerical illustrations, seeks another domain for
mathematical induction, or wishes to address open computational questions for an
honors class, there is much more that can be said.
To establish equation (34) begin with the first few values of such a sum as
n increases from unity. One can find these as 1, 5, 14, 30, and 55 by direct
computation and then notice each satisfies the right hand side of equation (34)
9
n
12
18
24
30
36
n+1
13
19
25
31
37
2n + 1
25
37
49
61
73
RHS of (34)
650
2109
4900
9455
16206
Table 5: Product for Squares Summed
for the corresponding n values, 1, 2, ..., 5. So since (34) is known to be true at
some value, it can be proven using mathematical induction [3], [4], beginning with
the established pattern that it is true at these five n values. That is, the five
numeric patterns suggest seeking proof by establishing the inductive step via this
relationship:
(n + 1)(n + 2)(2n + 3)
n(n + 1)(2n + 1)
+ (n + 1)2 =
(35)
6
6
Evaluating the expressions on each of the two sides of (35) by multiplying out
terms gives the same result. Hence (34) is true in general.
8
Conclusion
Mathematics comes from observations about patterns in numbers and nature.
This paper described such relationships, beginning with one that extends the
Pythagorean relationship common in school-mathematics. The paper presented
the general form of relationships among sums of squares of consecutive integers. It
explained the starting values in such an ascending series by displaying a new notion
in pattern thinking. That notion was the pivot as a starting point for counting
down and up.
Theme pivot develops pattern thinking with respect to symmetry within a
number series. That approach was used to set up similar starting value relationships among sums of consecutive integers. Having resolved issues about squares
and first-power sums involving consecutive integers we presented material about
representing cubed positive integers as sums of adjacent odd numbers.
These three topics relate to a conjecture mathematics has considered to be
very important [14], because they ask questions about sums of numbers raised to
different exponents.
The final section concerned a change of focus from consecutive or adjacent values to summing squares beginning at unity. Material there covered three themes.
The first was in regard to the sum being square in sample numerical instances. The
10
second was the difference between instances/examples and existence/uniqueness.
Finally, this topic was used to address the central role of proof in mathematics
with particular reference to mathematical induction.
Throughout the material suggestions appear for class dialog and self-study.
9
Acknowledgement
Taylor B. Kang and Michelle M. Amagrande of California State Polytechnic University, Pomona, observed the relationships described in Values. Richard Korf,
University of California Los Angeles (UCLA), told me of the problem Watson
solved. David G. Cantor, UCLA and Center for Communications Research, San
Diego, suggested documenting Squares. The referee suggested that this material
had, through the range of exponents, a relationship to Fermat’s conjecture.
10
Table
Table
Table
Table
Table
Captions
1.
2.
3.
4.
5.
Consecutive Squared-Integer Sum Relationships
Cubes as Sums of Consecutive Odd Numbers
Product Pattern
Symmetry Pattern
Product for Squares Summed
References
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[2] Brain ticklers. The Bent of Tau Beta Pi, Spring 2004, p. 44; http://www.
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Mathematical Induction from Wikipedia.
wikipedia.org/wiki/Mathematical\_induction , 2007.
[4] Alexander Bogomolny.
Mathematical Induction.
cut-the-knot.org/induction.shtml , 2007.
http://en.
http://www.
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wolfram.comPythagoreanTriple.html , 1999.
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[6] E. W. Weisstein. Square Pyramidal Number. MathWorld; http://
mathworld.wolfram.com/SquarePyramidalNumber.html , 1999.
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pages 1–22, 1918.
[8] Keith J. Devlin. Mathematics: The Science of Patterns. W. H. Freeman and
Co., NY, 1994.
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cubes.jpg, 1997.
[10] Allen Klinger. A Cubed Positive Integer is a Sum of Consecutive Odd Numbers. http://www.cs.ucla.edu/~klinger/cubvis_6_15_01.htm, 2001.
[11] Martin Gardner. The Numerology of Dr. Matrix. Simon and Schuster, NY,
1967.
[12] Thomas L. Naps. Introduction to Data Structures and Algorithm Analysis,
2nd Edition, p. 50, prob. 4. West Publishing Co., St. Paul, Minnesota, 1992.
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[14] J. J. O’Connor and E. F. Robertson. History Topic Fermat. http://www.
google.com/search , 2007.
[15] Roger B. Nelson. Proofs Without Words Exercises in Visual Thinking. The
Mathematical Association of America, 1993.
[16] Russell Linton. Unpublished letter to Martin Gardner, 1967.
[17] T. H. Beldon. Runs of Squares. The Mathematical Gazette, pages 334–335,
December 1961.
[18] William Dunham. Journey Through Genius: The Great Theorems of Mathematics. John Wiley & Sons, Inc., NY, 1990.
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