Project Report
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MTH 496
Math 496 Project Report
Problem #14
John, Tom, and Ron
Professor Curtis Bennett
December 4, 2000
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Introduction: Fibonacci Numbers and the Euclidean
Algorithm
Problem #14: The smallest pair of positive integers so that the Euclidean Algorithm
when applied to them takes only one step is [1,1]. Similarly, the smallest pair of positive
integers so that the Euclidean Algorithm takes only two steps is [3,2].
(Step 1: 3 = 2 · 1 + 1, Step 2: 2=1·2 + 0).
Find the smallest pair of integers such that the Euclidean Algorithm takes three steps,
four steps, etc. State and prove a theorem for the case of n steps. Given a pair of two
positive integers, approximate a bound for how many steps the Euclidean Algorithm will
take.
The Euclidean Algorithm is a very useful algorithm for finding the greatest
common divisor of any two given integers. Using the Euclidean Algorithm is a tedious
task for pairs of large numbers because the algorithm requires a high number of steps to
execute. We figured that if we could approximate the number of steps, and furthermore
bound the number of steps then we could easily decide how practical the Euclidean
Algorithm would be for the given pair of numbers.
The Euclidean Algorithm is defined as follows1:
Let a and b be positive integers with b ≥ a. If a divides b, then the greatest
common divisor of a and b is a. If a does not divide b, then apply the Division Algorithm
repeatedly as follows:
aq0 + r0 = b
0 < r0 < a
r0q1 + r1 = a 0 < r1 < r0 …
The process ends when a remainder of 0 is obtained. This must occur after a finite
number of steps; that is, for some integer t:
rt-1qt + rt = rt-2
rtqt+1 + 0 = rt-1
0 < rt < rt-1
Then rt, the last nonzero remainder, is the greatest common divisor of a and b.
We began our investigation by looking at specific cases involving small integers.
When working through the different cases, we noticed that there might be a relation to the
Fibonacci sequence, which is defined as the sequence such that a n  a n 1  a n  2 . We did
not have anything specific, but we kept noticing the numbers of that sequence being ones
of interest. We noticed that when performing the Euclidean Algorithm on the Fibonacci
numbers we seemed to get, after a few steps, down to the cases we were given (that [1,1]
is the smallest pair that takes 1 step and that [3,2] is the smallest pair that takes two
1
This definition is taken from the Hungerford Abstract Algebra Text, by Thomas Hungerford, 1997,
Saunders College Publishing.
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steps). We then noted that the smallest pair that would take three steps would have to
have three as the second number and two as it’s first remainder thus setting it up to take
one more step than the case of [3,2]. So, if this new number gives a remainder of two
when performing the Euclidean Algorithm on [a3, 3], then the number a 3  3q 0  2 in
order to make a 3 the smallest integer, q 0 would have to equal one. So, a 3  3  2 or
a 0  5 . So the smallest pair that would take 3 steps is [5,3]. Not surprisingly, five is the
next Fibonacci number after three. We did more examples using the same idea and found
the smallest pair that would take four steps is [8,5] and the smallest for five steps is
[13,8]. These are these first seven numbers in the Fibonacci sequence. This led us to state
our theorem.
Defining “Smallest” for RON Pairs
Until this point we have been dealing with our problem almost entirely from an intuitive
standpoint, and a fairly ambiguous one at that. When we use the words “smallest” with
reference to a pair of integers, what precisely is meant? Obviously, we are talking not
just about a property of a single pair of integers, but a relation between two pairs; one
pair of integers is “smaller than” another pair. When comparing two integers (rather than
two integer pairs), a relationship involving a conception of magnitude is already defined:
namely, the “greater than” and “less than” relations, denoted symbolically by the symbols
“<” and “>”. The integers are an ordered set. However, no such ordering has been
previously defined for integer pairs.
While many number sets have such an ordering, including most familiar ones
(e.g. natural numbers, integers, rational numbers, and the real numbers), not every set of
numbers is ordered; for example, the complex numbers have no immediate ordering
principle—2i cannot be said to be less than 3i in the same way that 2 is less than 3.
However, complex numbers can be assigned a real number magnitude in the complex
plane that in principle could be used to order the set, at least partially. This has direct
bearing upon our Euclidean Algorithm integer pairs, since complex numbers involve
“pairs” of real numbers – a real coefficient and an imaginary coefficient. One could
choose to “order” the complex numbers based solely on one of these coefficients (the real
or the imaginary), or any number of other possibilities. Another analogous situation in
geometry is the measurement of perimeter versus area, or surface area as opposed to
volume; shapes can be related to one another based on two different computations of
magnitude.
Each potential ordering would yield different information about a number and its
relationship with other numbers, which could be used for different purposes. One could
imagine a plurality of such orderings, depending upon the task required for each.
What we want to do is to mathematically define our “smaller than” relationship
for integer pairs so that our ordering yields information germane to the task at hand:
namely, trying to determine the number of steps the Euclidean Algorithm takes to
compute the greatest common divisor of an arbitrary pair of positive integers. Ideally,
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our definition will reveal our conjecture about the Fibonacci numbers to be true. At the
same time we want to retain at least in part the intuitive conception of “smaller than” that
generated this conjecture. Logically, the relation “smaller than” could be defined in
almost any conceivable manner, up to and including the exact contrary of our intuitive
idea of “smaller,” or something almost entirely unrelated.
This is pure and original mathematics. The business of mathematicians initially
and for the most part is to frame precise concepts corresponding to our intuitive
apprehension. From this definitional (or axiomatic) foundation, we then employ
similarly precise argumentation to yield original knowledge in the form of logically
grounded results. When mathematics is at its best, not only are the axiomatic foundations
and definitional supplements clear and the system itself sound, but also new nonintuitive
knowledge is gained from these foundations. This affirms the fundamentally creative and
synthetic nature of mathematics while maintaining the importance of analytic thought and
reasoning from foundational security. From a student’s perspective, making such
conceptual definitions and subsequently proving theorems using those definitions is to
not only gain a crucial insight into how mathematics functions beyond the level of the
algorithm but to genuinely make a contribution to the body of mathematical knowledge
through the exploration of such definitions and their consequences.
Upon examination, it becomes apparent that there are many mathematical
interpretations for “smaller than,” each of which possesses a degree of intuitive validity.
Regardless of our definition, we certainly want to avoid blatantly counterintuitive
conclusions, both for the purposes of our proof and as a general heuristic principle. Thus,
for example, [1,2] should be smaller than [1000,1500]. It is clear that our task is greatly
simplified if we rely on the less than and greater than relations for the integers, our model
for an ordering relation. Each of the examples cited above in some way involve reducing
a question of n constants (e.g., length, width, and height) to a question of a single
constant (e.g., volume) and then comparing objects using less than, greater than, and
equality. Similarly, we want to derive some single number from each pair for the
purposes of comparison. Ostensibly, several examples immediately present themselves
as sound candidates. One could take the sum of the two integers, or, alternatively, the
product. Following geometry and analysis, one could consider the Euclidean norm of a
pair of integers; for example, the norm of [3,4] would be
32  42  5.
A wealth of other calculation of norms is available from other branches of
mathematics. However, certain cases present problems for these interpretations, which
demand consultation of our intuitive understanding of “smaller than” in the context of the
Euclidean Algorithm. Should [4,5] be smaller than [3,6]? The two pairs possess the
same sum. The product of [4,5] is greater than the product of [3,6]. Meanwhile the norm
of [4,5] is less than the norm of [3,6]. Clearly, then, these definitions are not equivalent.
Furthermore, upon inspection they are absolutely false on their face at predicting the
number of steps required to compute the greatest common divisor of a pair using the
Euclidean Algorithm. So what definition of “smaller than” provides the most pertinent
information for the pair’s greatest common divisor?
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Examining the Euclidean Algorithm, we take note of several facts. First, with
respect to the Euclidean Algorithm and greatest common divisors, the pairs themselves
are unordered; that is, the pair [a,b] is essentially equivalent to the pair [b,a] where a and
b are integers. This is not the case with all pairs of integers, even where the norms are
identical; for example, with points in the plane, or lengths and widths of rectangles, the
order of the numbers given yields very specific information about the mathematical
object being studied. Until this point, we have been speaking very loosely about a pair of
integers, and relations between those pair of integers, without defining precisely what we
mean by such a pair. This is crucial not only for greater precision in our reasoning, but
also, as we shall see, our “smaller than” ordering relation may well yield different
information for equivalent pairs if we do not clarify what we mean by a pair. For
example, under some definitions, “smaller than” could paradoxically state that the pair
[3,5] is smaller than the pair [5,3]; two equivalent pairs with respect to their GCD’s. In
another context, such definitions may actually be highly useful; but for the problem at
hand we want to be certain that our pairs or the “smaller than” relation are defined in
such a way that equivalent pairs are seen as such. Until we are ready to make these
definitions, however, we will by necessity continue to speak loosely about a “pair” of
integers.
Continuing, we note that in the second step of the Euclidean Algorithm, the lesser
integer of the pair is retained while the greater integer is discarded. For example, when
applying the Euclidean Algorithm on the pair [7,59], while the first step retains the
number 59, by the second step, it is discarded:
7 · 8 + 3 = 59
3·2+1=7
With the second step, a new Euclidean Algorithm begins, and similarly for each
subsequent step until a remainder of zero is reached – in this case, the Euclidean
Algorithm is essentially used after the second step to determine the GCD of the pair [3,7]
— by almost any intuitive definition a smaller pair than [7,59]. If we knew the number
of steps it took to compute the GCD of [3,7], we could simply add 1 (for the first step) to
find the number of steps needed to compute the GCD of [7,59]. Since the remainder
(which is carried over to the subsequent step) is always bounded above by the lesser
integer of the original pair, the larger number is simply irrelevant for determining a top
bound for steps of the Euclidean Algorithm. As the greater number grows larger and
larger, it only cycles modulo the lesser number with respect to its remainder; apart from
that, its size signifies nothing, since from its magnitude alone (in isolation from the other
number) we cannot determine what its remainder will be. Thus we see that the bound for
the number of steps of the Euclidean Algorithm can be determined completely from the
smaller number with greater precision before actually performing the algorithm than by
considering the two together in some combination. Hence in most cases, we want to
examine the lesser integer of the pair in order to determine whether or not one pair is
“smaller than” another. The intuitive exception would be in cases where the lesser
numbers in each pair were equal to one another; for example, ideally I would want [8,9]
to be smaller than [8,13], primarily because this satisfies my intuition that since 9 < 13,
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[8,9] would be smaller than [8,13], but also since [8,9] takes fewer steps under the
Euclidean Algorithm than [8,13]. So, since the Euclidean Algorithm depends almost
entirely on the lesser integer of the pair, we will define our pairs and our relation
accordingly.
We have done the exploratory work needed to determine what we have meant in
our casual discourse about pairs of integers and the “smaller than” relation and properly
motivate their definitions. Now we are ready to precisely define these terms and to use
them to prove some important facts about the Euclidean Algorithm2.
--(1) Definition: A RON Pair is any pair of positive integers [a,b] where a < b.
Consequently the set of all RON Pairs is the set S = {(a,b) : a,b  N, b – a  N }, taking
N to be the set of positive integers (assumed here to be known or constructed), and b – a
 N to be the customary definition of a < b. Observe that for every ordered pair of
positive integers (a,b) in N2 , excepting pairs of the form (a,a), there is a corresponding
RON Pair—either [a,b] (if a < b ) or [b,a] (if b < a). We are justified in excluding pairs
of the form (a,a) since the Euclidean Algorithm on such pairs is essentially redundant;
their GCD is automatically a.
(2) Definition: For any two RON Pairs [a,b] and [c,d], [a,b] is said to be smaller than
[c,d] if one of the following mutually exclusive conditions hold:
1) a < c ;
2) a = c and c < d.
In such cases we write [a,b] ◄ [c,d]. In such a case [c,d] is said to be bigger than [a,b].
If a = c and c = d then [a,b] is said to be equal to [c,d], written [a,b] = [c,d].
In a set W of RON pairs, [a,b] is said to be the smallest pair in W if for any other RON
pair [e,f] in W, [a,b] ◄ [e,f].
--Now we observe some facts about the ◄ , or smaller than, relation. We have chosen this
symbol primarily to suggest the symbol for the less than relation (<), and many of the
following facts rely primarily on facts about the < relation. This is an excellent
opportunity to concretely explore some important logical relations seen in algebra, such
as transitivity, symmetry, etc. This also foreshadows some important concepts in higher
mathematics while reinforcing our apprehension of their presence in logical relations
used every day such as the equality relation and the less than relation.
(3) Trichotomy. For any two RON Pairs [a,b] and [c,d], exactly one of the following
holds:
1) [a,b] ◄ [c,d];
2) [a,b] is bigger than [c,d];
3) [a,b] = [c,d].
This follows directly from trichotomy in the integers. Suppose [a,b] and [c,d] are RON
pairs. Then a, b, c, d are all positive integers. Hence a < c, a > c, or a = c, and b < d, b
2
Furthermore, since we have demonstrated the exhaustive mathematical work necessary to motivate these
definitions, proving our mettle as mathematicians, we are now justified in naming a few things after
ourselves.
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> d, or b = d. If a < c, then [a,b] ◄ [c,d] by (2). If a > c, then [a,b] is bigger than [c,d].
If a = c, then we examine b and d. If b < d, then [a,b] ◄ [c,d]. If
b > d, then [a,b] is bigger than [c,d]. If b = d, then [a,b] = [c,d]. Since this exhausts all
possible cases, our proposition is proven.
(4) The relation “equal to” is an equivalence relation; i.e., it is reflexive, symmetric, and
transitive. This proof similarly follows directly from the properties of the equality
relation on the integers and is trivial.
(5) The smaller than relation (◄) is irreflexive and antisymmetric. Since [a,b] = [a,b],
[a,b] is never smaller than [a,b] by trichotomy. Similarly, if [a,b] ◄ [c,d], then [c,d] is
greater than [a,b] and consequently [c,d] cannot be less than [a,b].
(6) ◄ is transitive. Consequently if [a,b] ◄ [c,d] and [c,d] ◄ [e,f], then [a,b] ◄ [e,f].
Suppose that [a,b] ◄ [c,d] and [c,d] ◄ [e,f]. Then a < c or a = c and b < d, and c < e or
c = e and d < f. Thus we distinguish four cases and proceed as follows:
Case One: a < c and c < e. Then since < is transitive, a < e and thus [a,b] ◄ [e,f].
Case Two: a < c, c = e and d < f. Then a < c = e implies that a < e and thus
[a,b] ◄ [e,f]
Case Three: a = c, b < d, and c < e. Then a = c < e implies that a < e and thus
[a,b] ◄ [e,f]
Case Four: a = c, b < d, c = e, d < f. Then a = c = e implies that a = e. Also b < d,
d < f implies that b < f by transitivity. Thus [a,b] ◄ [e,f].
Since this exhausts all possible cases, ◄ is transitive.
Fibonacci Numbers and the JOHN Theorem
In the previous section, we made some important definitions that now enable us to deal
with an ordering for pairs of positive integers. From this point we made the natural move
to examine the properties of the smaller than relation for the set of RON pairs. We could
easily extend these definitions further, defining RON n-tuples in the same lexicographic
style, and extending the definition of smaller than accordingly. This pure mathematics
could conceivably have a number of applications. However, at this point we want to turn
to the particular “application” that motivated our definition of both RON pairs and the
smaller than relation: the Euclidean Algorithm.
(7) The JOHN Theorem for the Euclidean Algorithm. Let F(k) be the kth number in
the Fibonacci Sequence, where F(0) = 1, F(1) = 1, and F(k+1) = F(k) + F(k-1). Then for
n ≥ 2, [F(n), F(n+1)] is the smallest RON pair for which the Euclidean Algorithm takes n
steps. –
Since the Fibonacci numbers are defined recursively, it seems natural to prove
this theorem by induction on n. First, however, we want to observe the following fact
about the Euclidean Algorithm on [F(k), F(k+1)]. Since F(k+1) = F(k) + F(k-1) for all k
> 1, the Euclidean Algorithm proceeds as follows:
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F(k) · 1 + F(k-1) = F(k+1)
F(k-1) · 1 + F(k-2) = F(k)
F(k-2) · 1 + F(k-3) = F(k-1)…
F(1) · 1 + F(0) = F(2)
F(0) · 1 + 0 = F(1).
Thus every step in the Euclidean Algorithm involving consecutive Fibonacci numbers
involves more Fibonacci numbers—in fact, every Fibonacci number less than or equal to
F(k+1). Also, every Fibonacci number is relatively prime to its successor and its
predecessor, since the last nonzero remainder is always F(0) = 1. We will use the first
observation in our proof repeatedly, but both observations give us some indication as to
why the Fibonacci numbers have this particular relationship with the Euclidean
Algorithm.
Now that we possess the proper tools in terms of our definitions, the proof is
disarmingly simple.
Proof: By induction on n. For n = 2. [2,3] is the smallest pair of integers for which the
Euclidean Algorithm takes 2 steps, and 2 is the 2nd Fibonacci number while 3 is the
third. Suppose that for all integers k ≥ 2 and less than or equal to n, [F(k-1), F(k)] is the
smallest pair of integers such that the Euclidean Algorithm takes k-1 steps.
We want to show that [F(n), F(n+1)] is the smallest pair of integers such that the
Euclidean Algorithm takes n steps.
(8) Claim 1: The Euclidean Algorithm on [F(n), F(n+1)] takes n steps.
Proof: Since we are dealing with the Fibonacci sequence, by definition
F(n+1) = F(n) · 1 + F(n-1), where F(n-1) < F(n). (Note: This fact proves that
[F(k), F(k+1)] is a RON pair for all k ≥ 1.) Thus the second step of the Euclidean
Algorithm is F(n) = F(n-1) · 1 + F(n-2). But by our hypothesis [F(n-1), F(n)] takes n - 1
steps. Thus the Euclidean Algorithm terminates after n - 1 more steps. Therefore, adding
only our first equation, we see that [F(n), F(n+1)] takes a total of n steps.
(9) Claim 2: [F(n), F(n+1)] is the smallest pair of integers with n steps.
Proof: Let [a,b] be a RON pair smaller than [F(n), F(n+1)]. Then a < F(n) or
a = F(n) and b < F(n+1). We want to show that the Euclidean Algorithm on [a,b] takes
less than n steps. Consider the Euclidean Algorithm on [a,b]. By the Division Algorithm
there exist integers q0 and r0 such that
(9.0) b = aq0 + r0, where 0 ≤ r0 < a.
Now the second step of the Euclidean Algorithm (if it continues) is
(9.1) a = r0q1 + r1, where q1 is an integer and 0 ≤ r1 < r0. The third step (if
it continues) is
(9.2) r0 = r1q2 + r2.
Case One: a < F(n).
Case 1a: r0 ≤ F(n-1). Then [r0, a] is smaller than [F(n-1), F(n)] and therefore takes
less than n - 1 steps. Thus, adding (9.0) as the first step, we see that [a,b] takes
less than n steps.
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Case 1b: r0 > F(n-1). By (9.0) a > r0, such that a = r0 + some positive integer. Therefore
q1 ≥ 1. We will show that r1 < F(n-2). Suppose to the contrary that r1 ≥ F(n-2).
Then by (1) a = r0q1 + r1 > F(n-1) + F(n-2) = F(n), such that a > F(n). But this
contradicts our assumption that a < F(k). Therefore r1 < F(n-2). Thus [r1, r0] is
smaller than [F(n-2), F(n-1)]. Hence by our induction hypothesis the Euclidean
Algorithm on [r1,r0] takes less than n - 2 steps. Adding equations (9.0) and (9.1)
as our first and second steps, respectively, we see that [a,b] must take less than n
steps.
Case Two: a = F(n) and b < F(n+1). Then by substitution
F(n) + r0 = a + r0 = b < F(n+1) = F(n) + F(n-1).
Hence r0 < F(n-1). Therefore [r0, a] is smaller than [F(n-1), F(n)] and hence the
Euclidean Algorithm takes less than n -1 steps. Thus the Euclidean Algorithm on
[a,b] takes less than n steps.
Therefore [F(n), F(n+1)] is the smallest pair of integers for which the Euclidean
Algorithm takes n steps. QED.
Approximating a Bound – The TOM Algorithm
With the previous theorem, we have shown that the smallest RON pair for which the
Euclidean Algorithm takes n steps is given by the Fibonacci Numbers [F(n), F(n+1)].
Thus conceivably for any RON pair [a,b], we can bound the number of steps the
Euclidean Algorithm takes on [a,b] by the next highest pair of Fibonacci numbers. Since
there are infinitely many Fibonacci numbers, for every integer a, there exists a Fibonacci
number F(n) such that a < F(n). By the theorem just proven, if a < F(n), then [a,b] is
smaller than [F(n), F(n+1)] and thus for any integer b > a, computing the GCD of [a,b]
takes less than n steps. If a = F(n), then a takes n or fewer steps: if b < F(n+1), we can
guarantee that the Euclidean Algorithm takes less than n steps; if b > F(n+1), it may take
up to n steps, and if b=F(n+1), it takes precisely n steps.
Can we use the theorem just proven to find a simple computation to bound the
number of steps the Euclidean Algorithm takes for an arbitrary RON Pair? Upon
examination of the first n Fibonacci numbers for low n, we notice a pattern. There are no
more than five Fibonacci numbers with m digits for any positive integer m except for the
first 6 Fibonacci numbers, which each have 1 digit. If we could guarantee that there were
five Fibonacci numbers for every number of digits m, we could easily determine a bound
for the Euclidean Algorithm for [a,b] by finding the number of digits of a and then
finding the largest Fibonacci number with the same number of digits. If a has the same
number of digits as F(n), where F(n) is the largest such Fibonacci number, then a is
clearly less than F(n+1), since F(n+1) has more digits than a. Hence [a,b] is smaller than
[F(n+1), F(n+2)] and consequently the Euclidean Algorithm on [a,b] takes n or fewer
steps.
Upon further observation, it becomes clear that not every set of numbers with m
digits contains 5 Fibonacci Numbers. Some contain 4. This is not a damaging
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consequence, however, since we want to approximate an upper bound. If we were to
assume that every such set did contain 5 Fibonacci Numbers, we will actually be
overestimating our guess for n, where F(n) is the largest Fibonacci number with m digits.
We only need to make sure that there can be no more than 5 Fibonacci numbers with m
digits for any m.
(10) Lemma: For all Fibonacci numbers F(n), 2F(n) ≥ F(n+1).
Proof: Now for n = 0, clearly 2 ≥ 1. For n > 0, F(n + 1) = F(n) + F(n-1). Since
F(n) ≥ F(n-1), F(n+1) ≤ F(n) + F(n) = 2F(n).
(11) TOM’s Proposition: For any positive integer m, the largest possible Fibonacci
number with m digits is less than or equal to F(5m). The proof is by induction on m.
First note that the largest Fibonacci number with 1 digit is F(5) = 8. Now assume that for
all positive integers k where k ≤ m, the largest Fibonacci number with k digits is less than
or equal to F(5k). We want to show that the largest Fibonacci number with m + 1 digits
is less than or equal to F(5(m+1)). Let f1 be the smallest Fibonacci number with m + 1
digits. Then f1 ≥ 10m+1 , since 10m+1 is the smallest integer with m + 1 digits.
Consequently f0, the Fibonacci number immediately preceding f1, must be greater than or
equal to 5·10m. Hence f2 = f1 + f0 ≥ 10m+1 + 5m = 10m (15). Similarly, f5 = 3f2 + 2f1 ≥ 3 ·
10m(15) + 2 · 10m+1 = 10m(65), and f4 ≥ 10m (40). Consequently f6 = f5 + f4 ≥ 10m(65) +
10m(40) = 10m(105) = 10m+2 +10m(5). Hence f6 has at least m + 2 digits. So all Fibonacci
numbers with m + 1 digits or fewer are less than or equal to f5. Since by our hypothesis
f0 ≤ f(5m), f5 ≤ f(5m+5) = f(5(m+1)). QED.
How can we determine the number of digits of an arbitrary positive integer a?
This is easy enough; simply take the integer part of the common logarithm function. This
in conjunction with the proposition just proven yields the following algorithm for
bounding the number of steps the Euclidean Algorithm takes to find the GCD on an
arbitrary RON pair [a,b]:
(12) TOM’s Algorithm: Let [a,b] be any RON pair. The number of steps for the
Euclidean Algorithm on [a,b] is bounded above by 5 · Ipart(log10 a), where Ipart(x) is the
greatest integer less than or equal to x. This follows directly from the JOHN Theorem
and TOM’s Proposition.
Refining the bound
Our second approximation is similar, except it uses the Fibonacci number just greater
than or equal to
a - Ipart (a/b) ∙ b.
This is the first remainder obtained from the first step of the Euclidean Algorithm
performed on the pair [a, b]. This provides for a better bound than the smallest number of
[a,b] because it accounts for the possibility of the numbers being very large, yet relatively
close. For example, suppose we wanted to estimate the number of steps required to
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perform the Euclidean Algorithm on the pair [38,43]. From our first method we would
have gotten that rounding the lower number up to the next Fibonacci number gets us to
55. Since 55 = F(9) from our definition of the Fibonacci numbers earlier, we would
estimate it to take nine steps. However, taking the first remainder of this division (5),
since 5 = F(4), adding one to account for the first step taken to find this remainder, our
new estimate would be 5 steps. This is a much smaller and more precise upper bound
than our initial estimate.
Our third approximation is to take successive remainders from [a,b] using the
Euclidean Algorithm. Using this, we can get a better approximation when [a,b] are large
integers, approximately anything over 100 -500. After each step, the RON pairs being
studied become smaller and smaller, and consequently the estimate for the bound
becomes more and more precise. Furthermore, large pairs with trivial early remainders
are weeded out. However, with exceedingly large numbers, the remainders become more
and more difficult to calculate. Large numbers will always pose a significant problem for
any bound approximation.
Examining a logarithmic graph of the Fibonacci numbers (see figure 1 below), it
becomes clear that a better logarithmic approximation than the common logarithm
probably exists. Our method for the common logarithm is very useful for finding rough
bounds for relatively smaller pairs, particularly when the number of digits can simply be
observed without computing the logarithm. However, it has a significant drawback: for
larger pairs of numbers it increasingly overestimates the number of steps needed. Further
exploration with the Fibonacci numbers is required in order to determine a better bound.
This is a natural expansion of the work begun here.
Everything here, apart from some of the language and imported apparatus of
mathematical proofs, relies solely upon knowledge and computations that are possible at
the high school level. However, the themes raised by this exploration touch on many
important themes in higher level mathematics that could be explored for the first time in a
high school setting: the idea of recursive definition, the importance and creativity
involved in precise definition of concepts, and the principle of mathematical induction.
Students are also introduced to two important and ubiquitous ideas in mathematics: the
Euclidean Algorithm and the Fibonacci Sequence. Students can see how seemingly
unrelated mathematical ideas can involve one another in interesting and unexpected
ways, which then can be used to predict information regarding the greatest common
divisor, among other applications.
Also, we have laid the groundwork for further exploration when more mathematics can
be utilized, such as the generating function for the Fibonacci sequence, which can be used
to derive a more accurate bound. But this is beyond the scope of what we have done
here.
----
-- Page 11 --
Project Report
2/6/16
MTH 496
1E+16
1E+14
1E+12
1E+10
1E+08
1E+06
10000
100
1
64
55
46
37
28
19
10
Series1
1
F(n)
Fibonacci numbers
Cell number(n)
Figure 1. Fibonacci numbers vs. logarithmic graph of the value of the Fibonacci number.
-- Page 12 --
Project Report
2/6/16
MTH 496
Weekly Journal for:
JOHN, TOM, and RON
Week 1
We got project #14. We did a multitude of plugging and chugging different integers into
the Euclidean Algorithm to find the number of steps it took to find the GCD. We saw
that the Fibonacci numbers appeared to be related somehow to the solution of the
problem.
Week 2
We are sure that for each increase in the number of steps of the Euclidean Algorithm that
the pairs of subsequent Fibonacci numbers provide the smallest integers for that number
of steps. We want to prove this with an induction proof. We are conjecturing that the
smaller integer of a pair of successive Fibonacci numbers is a bound for the number of
steps the Euclidean Algorithm would take to find the GCD. Work has been started to
develop a spreadsheet on Excel to find the GCD.
Week 3
We are wondering what the smallest integer means. We realize we can define this at
least three different ways. We develop a pre-condition for the problem, that is, we want
to be able to distinguish between the pairs (1,1) and (2,1) since both pairs take 1 step
each.
Week 4
We did not meet as a team this week. We made an Excel workbook to help with our
investigations, and hopefully will be used to demonstrate our project to others. The
spreadsheets calculate the greatest common divisor (GCD) of two integers, and produce a
Fibonacci number corresponding to integer input from one to seventy. We developed a
draft theorem which is, "The smallest pair of positive integers (a,b), such that the
Euclidean Algorithm takes n steps is the two successive terms of the Fibonacci sequence
(F(n+1), F(n)), with n greater than or equal to 2, and F(0) = F(1) = 1."
Week 5
We all worked through out the week on our own. We came up with an induction proof
for our function. We also came up with a new bound for the case of n steps, that the
number of steps for the Euclidean Algorithm is the number of steps that corresponds to
the Fibonacci number that is just less than or equal to the remainder when subtracting the
original integers. It is not proven. We also noticed that the limit of the function
F(n)/F(n+1) is the golden ratio. We don’t know how this relates to our problem, but we
found it interesting.
Week 6
We met with the Professor this week. We decided that the definition of smallest we will
use for our proof is the number with the smallest magnitude of a pair of positive integers.
-- Page 13 --
Project Report
2/6/16
MTH 496
Week 7
We outlined the proof of our theorem. The proof is by induction as originally surmised.
Week 8
We decided to write a GCD program using the TI-83 because of programming difficulties
in Excel. We started the project write up, and made progress with the TI 83 Plus
program. The final write up is shaping up as a chronological report, since the chronology
corresponds with the proof of the problem. We also have come up with a few different
ideas of where we would like to go from here. We have also come up with an idea of
using the Euclidean Algorithm for finding the GCD of a larger set of numbers. We have
also talked about defining the properties of lexicographical ordering of pairs.
Week 9
We worked for a short time on writing up the final report on what we already have. We
looked at the case of n-tuples and tried different conjectures on the number of steps it
would take to find the GCD of an n-tuple. We looked at two different methods of finding
the GCD for a set of numbers greater than two. The first way is by finding the GCD of
every successive pair of numbers and the second is by taking the smallest number and
finding the GCD of that and every other number in the set. This work intertwined with
our approximation of a bound for the number of steps the Euclidean Algorithm will take
to find the GCD of a pair of integers. We questioned that our bound isn’t close as other
options for large numbers. Thus we observed that there are no more than five Fibonacci
numbers for any set of numbers within a given length of digits. We tried developing a
formula for the relationship between the number of digits of an integer (assuming it is the
smallest integer of a set) and the number of steps it would take to use the Euclidean
Algorithm to find the GCD of the given set. We didn’t come up with that, but did come
up with a proof of our observation. This led us to see that a way of finding a Fibonacci
number for any given integer can be done by knowing the number of digits in the given
number. We continued examining our approximation for a bound and looked at cases
with larger numbers and saw that our approximation works better each time we perform
the Euclidean Algorithm.
Week 10
We made a graph of the first 70 Fibonacci numbers using a logarithmic table in Excel,
did some research in the library, and worked with a generating function we found in a
book about Fibonacci numbers. This is the first time we used the library for our project.
We did this because we felt we had hit a stage where we wanted to answer some
questions for ourselves and knew that we couldn’t impact the project at this point. We
also made progress in developing a TI-83 program to give the GCD of two small integers.
We are working on coming up with an approximation for the bound based on the graph of
the Fibonacci sequence. We need to start dedicating ourselves to staying focused on
writing the report or making a presentation instead of working on different aspects of the
problem. If we could continue forging forward with exploring the problem, then we could
look at how Fibonacci numbers relate to decimal expansions, and what role the
logarithmic graph plays in applications.
-- Page 14 --
Project Report
2/6/16
MTH 496
ATTACHMENT #1
TI - 83 Plus Program
Calculating the GCD and # of steps the
Euclidean Algorithm takes to find the GCD
PROGRAM: GCDLCT
:Prompt A,B
:ClrList L1
:ClrList L2
:100dim(L1)
:100dim(L2 )
:
:For(X,1,100)
:iPart(A/B)Q
:QL2 (X)
:A - (Q*B)L1 (X)
:BA
:If L1 (X) = 0
:Goto GC
:L1 (X)B
:End
:
:Lbl GC
:If X=1
:Then
:Output(5,1,”GCD= “)
:Output(5,8,B)
:1V
:Goto GD
:Else
:X-1V
:End
:
:Output(5,1,”GCD= “)
:Output(5,8,L (V):L )
:
:Lbl GD
:Output(6,1,”NUM OF STEPS “)
:Output(6,14,V)
-- Page 15 --
Project Report
2/6/16
MTH 496
ATTACHMENT #2
EXCEL Worksheet: Finding the GCD
Given (a,b) find the greatest common
divisor
Euclidean Algorithm: a = bq + r
type any positive
Type in a 2nd
integer
(+)integer
146
arrange arrange INT(a/b) b*q
finding r
da
db
144
146
144
1
144
2
144
2
72
144
0
144
0 #DIV/0! #DIV/0! #DIV/0!
-- Page 16 --
Project Report
2/6/16
MTH 496
ATTACHMENT #3
The First 70 Fibonacci Numbers
Finding the Fibonacci Number
Type in n
between
1 and 70:
N
3
F(n)
3
0
1
2
1
1
2
3
4
5
6
3
5
8
13
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
75025
121393
196418
317811
514229
832040
1346269
2178309
3524578
5702887
9227465
=
-- Page 17 --
Project Report
2/6/16
MTH 496
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
2971215073
4807526976
7778742049
12586269025
20365011074
32951280099
53316291173
86267571272
1.39584E+11
2.25851E+11
3.65435E+11
5.91287E+11
9.56722E+11
1.54801E+12
2.50473E+12
4.05274E+12
6.55747E+12
1.06102E+13
1.71677E+13
2.77779E+13
4.49456E+13
7.27235E+13
1.17669E+14
1.90392E+14
3.08062E+14
-- Page 18 --