Senior Seminar Project
By Santiago Salazar
Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse
is equal to the sum of the squares of the legs.
The Pythagorean Proposition
by Elisha S. Loomis (1852 - 1940)
 A compilation of proofs to the Pythagorean Theorem
 It contains more than 367 distinct demonstrations
 The proofs were classified into four categories
 No trigonometric proofs since the whole body of
trigonometry depends on the Pythagorean Theorem,
and in particular, on the relationship
sin2 𝜃 + cos 2 𝜃 = 1, which is derived from it
 Loomis clarifies that a trigonometric proof constitutes
circular reasoning and is therefore incorrect
Four kinds of Demonstrations
 Those based upon linear relations –the algebraic
proofs
 Those based upon comparison of areas –the geometric
proofs
 Those based upon vector operations –the quaternionic
proofs
 Those based upon mass and velocity –the dynamic
proofs
Pythagoras (ca. 575-495 BCE)
 Born on the Island Samos, off the coast of modern-day
Turkey
 There is no reliable information about him
 Traveled through Egypt and Mesopotamia, where he
probably increased his knowledge of Mathematics,
Philosophy, and Religion
 Also traveled to Miletus, where he made advances in
geometry under philosophers and mathematicians
such as Thales of Miletus, Anaximander, and
Anaximenes
Pythagoras (ca. 575-495 BCE)
 Moved to Croton (today, Crotone in southern Italy),
about 530 BCE
 There he founded a society, denominated the
Pythagoreans, whose main interests were religion,
mathematics, astronomy, and music
 The nature of the universe could be explained by
numbers and their ratios
Egypt
 During his trips to Egypt, Pythagoras probably came in
contact with the measuring method of the
Harpedonapts (rope stretchers)
 Egyptian used, for architectural purposes, ropes tied
with 12 equidistant nodes to create right triangles of
lengths 3,4, and 5 units
 This suggests that the Egyptians had some insights
about the special case of the 3-4-5 right triangle well
before Pythagoras proved the more general version
Harpedonapts (Rope strechers)
Mesopotamia (Babylonians)
 Babylonians had some insights about this theorem ca.
1800 BCE
 This shows the high level of mathematical knowledge
that existed well before the Greeks
 They discovered a significant number of Pythagorean
triples
 They solved problems that could only be solved with
the use of the Pythagorean triples and the Theorem of
Pythagoras
Plimpton 322 (ca. 1800 BCE)
Plimpton 322 (ca. 1800 BCE)
 A Babylonian clay table containing Pythagorean triples
 G.A. Plimpton Collection at Columbia University
 One column is missing, which is believed to contain
the third number in each Pythagorean triple
 15 rows and 4 columns
India and China
 About 800 BCE in India, ancient mathematicians
solved problems that relate to the Pythagorean
relation
 Uses of the Pythagorean Theorem appeared in “Nine
Chapters on the Mathematical Arts,” probably the
most influential Chinese mathematical work
 The Chinese provided a proof of the Theorem only in
the special case of the 3-4-5 triangle
Definition
A Pythagorean triangle is a right-angled triangle
whose side’s lengths are positive integers.
Equivalent Problem
As a result of the Pythagorean Theorem and the previous
definition, our problem of constructing the integer
solutions to the equation 𝑥 2 + 𝑦 2 = 𝑧 2 is equivalent to
that of finding all Pythagorean triangles, whenever we
restrict ourselves to positive integer solutions only.
Definitions
 A triple is an element of the set ℝ × ℝ × ℝ =
𝑎, 𝑏, 𝑐 𝑎, 𝑏, 𝑐 ∈ ℝ
 Throughout, we let 𝑆 denote the set of positive integer
triples, i.e. 𝑆 = 𝑎, 𝑏, 𝑐 ∈ ℕ × ℕ × ℕ 𝑎2 + 𝑏 2 = 𝑐 2 .
Thus a positive integer triple is in 𝑆 if and only if it is
an integer solution to our initial equation 𝑥 2 + 𝑦 2 = 𝑧 2
 An element of 𝑆 is called a Pythagorean triple
 Thus a Pythagorean triple generates a Pythagorean
triangle, and conversely, a Pythagorean triangle
generates a Pythagorean triple
Analysis of the Problem (part I)
 As with any mathematical problem, we try to
transform our current problem into a simpler one
 First we construct a “very rich” subset 𝑆′ of 𝑆
 Since 𝑆′ is “very rich,” it will allow us to recover 𝑆 in a
manner that will become apparent later
 But before doing this, we need to build some
machinery
Definition 1
We say that 𝑎 divides 𝑏 (denoted by 𝑎|𝑏) if and only if
there exists an integer 𝑑 such that 𝑏 = 𝑎𝑑.
Definition 2
We say that 𝑑 is greatest common divisor of 𝑎 and 𝑏
(denoted by 𝑑 = 𝑔𝑐𝑑 𝑎, 𝑏 ) if and only if
(i) 𝑑|𝑎 and 𝑑|𝑏, and
(ii) if 𝑐|𝑎 and 𝑐|𝑏, then 𝑐 ≤ 𝑑.
Definition 3
We say that 𝑎 and 𝑏 are relatively prime if and only if
𝑔𝑐𝑑 𝑎, 𝑏 = 1.
Theorem 4
If 𝑑 = 𝑔𝑐𝑑 𝑎, 𝑏 , then 𝑔𝑐𝑑 𝑎/𝑑, 𝑏/𝑑 = 1.
Proposition 5
If 𝑔𝑐𝑑 𝑎, 𝑏 = 1 and 𝑎2 + 𝑏 2 = 𝑐 2 , then 𝑔𝑐𝑑 𝑎, 𝑐 = 1
and 𝑔𝑐𝑑 𝑏, 𝑐 = 1.
Definition 6
We say that a positive integer solution 𝑎, 𝑏, 𝑐 to the
equation 𝑥 2 + 𝑦 2 = 𝑧 2 (i.e. a Pythagorean triple or
equivalently an element of 𝑆) in which 𝑔𝑐𝑑 𝑎, 𝑏 = 1 is a
fundamental solution.
Analysis of the Problem (part II)
 The first 6 results allow us to perform the desired
transformation
 Let 𝑆 ′ =
𝑎, 𝑏, 𝑐 ∈ ℕ × ℕ × ℕ 𝑎2 + 𝑏 2 = 𝑐 2 and 𝑔𝑐𝑑 𝑎, 𝑏 = 1
 Thus instead of trying to construct the set 𝑆, we try to
construct the set 𝑆′, which turns out to be much
simpler
 Yet, the set 𝑆′ is rich enough to allow us to recover the
set 𝑆 back
Analysis of the Problem (part II)
The recovery process is as follows:
 Let 𝑎, 𝑏, 𝑐 ∈ 𝑆
 Case 1: 𝑎, 𝑏, 𝑐 ∈ 𝑆′. Then we are “good”
 Case 2: 𝑎, 𝑏, 𝑐 ∈ 𝑆 − 𝑆′, then 𝑎2 + 𝑏 2 = 𝑐 2 and d =
𝑔𝑐𝑑 𝑎, 𝑏 > 1. In this case 𝑎/𝑑, 𝑏/𝑑, 𝑐/𝑑 ∈ 𝑆′
 This shows that any element of 𝑆 can be obtained from an
element in 𝑆′ if we multiply by a suitable factor
 More precisely, 𝑆 ⊆ 𝑑∈ℕ 𝑑 ∙ 𝑆 ′
 Since 𝑆 ′ ⊆ 𝑆 by construction, we obtain 𝑆 = 𝑑∈ℕ 𝑑 ∙ 𝑆 ′
Lemma 7
If 𝑎, 𝑏, 𝑐 is a fundamental solution, then exactly one of
𝑎 and 𝑏 is even and the other is odd.
Corollary 8
If 𝑎, 𝑏, 𝑐 is a fundamental solution, then 𝑐 is odd.
Lemma 9
If 𝑟 2 = 𝑠𝑡 and 𝑔𝑐𝑑 𝑠, 𝑡 = 1, then both 𝑠 and 𝑡 are
squares.
Lemma 10
Suppose that 𝑎, 𝑏, 𝑐 is a fundamental solution and 𝑎 is
even. Then there are positive integers 𝑚 and 𝑛 with 𝑚 >
𝑛, 𝑔𝑐𝑑 𝑚, 𝑛 = 1, and 𝑚 ≢ 𝑛 𝑚𝑜𝑑 2 such that
𝑎 = 2𝑚𝑛,
𝑏 = 𝑚2 − 𝑛2 ,
𝑐 = 𝑚2 + 𝑛2 .
Lemma 11
If
𝑎 = 2𝑚𝑛,
𝑏 = 𝑚2 − 𝑛2 ,
𝑐 = 𝑚2 + 𝑛2 ,
then 𝑎2 + 𝑏 2 = 𝑐 2 . If in addition, 𝑚 > 𝑛, 𝑚 and 𝑛 are
positive, 𝑔𝑐𝑑 𝑚, 𝑛 = 1, and 𝑚 ≢ 𝑛 𝑚𝑜𝑑 2 , then
𝑎, 𝑏, 𝑐 is a fundamental solution.
Theorem 12
𝑎, 𝑏, 𝑐 where 𝑎 is even is a fundamental solution if and
only if there are positive integers 𝑚 and 𝑛 with 𝑚 > 𝑛,
𝑔𝑐𝑑 𝑚, 𝑛 = 1, and 𝑚 ≢ 𝑛 𝑚𝑜𝑑 2 such that
𝑎 = 2𝑚𝑛,
𝑏 = 𝑚2 − 𝑛2 ,
𝑐 = 𝑚2 + 𝑛2 .
Consequences
 As a consequence of the previous theorem, we see
what are the necessary and sufficient conditions for a
Pythagorean triple to be a fundamental solution
 This allows us to construct a “very large” subset of 𝑆,
denoted by 𝑆′
 The elements of 𝑆 − 𝑆′ can be easily obtained from 𝑆′
 In fact, 𝑆 = 𝑑∈ℕ 𝑑 ∙ 𝑆′
Some Fundamental Solutions
𝒎
𝒏
𝒂
𝒃
𝒄
𝒂𝟐
𝒃𝟐
𝒄𝟐
2
1
4
3
5
16
9
25
3
2
12
5
13
144
25
169
4
1
8
15
17
64
225
289
4
3
24
7
25
576
49
625
5
2
20
21
29
400
441
841
5
4
40
9
41
1600
81
1681
6
1
12
35
37
144
1225
1369
7
2
28
45
53
784
2025
2809
Curiosities
 Interesting results are merely immediate consequences
of the Pythagorean Theorem such as the discovery of
the existence of irrational numbers
 The Pythagorean Theorem also has applications in
fractal art and music
 They are not easily deducted at first glance
 The construction of set 𝑆 of Pythagorean triples gives
us a way to easily prove an immense number of
curiosities, otherwise not possible, because this tells us
about the nature of 𝑎, 𝑏, and 𝑐 in a Pythagorean triple
𝑎, 𝑏, 𝑐
Curiosity 1
If 𝑎, 𝑏, 𝑐 ∈ 𝑆, then
 4|𝑎,
 3|𝑎 or 3|𝑏,
 12|𝑎𝑏,
 6|𝑎𝑏/2,
 5|𝑎 or 5|𝑏 or 5|𝑐,
 60|𝑎𝑏𝑐,
 𝑐 − 𝑎 𝑐 − 𝑏 /2 is a perfect square
Curiosity 2
 There is a variety of connections between the
Fibonacci sequence 1,1,2,3,5,8,13,21,34,55,89,144, …
and Pythagorean triples
 One of them is the generation of Pythagorean triples
from the Fibonacci sequence
 Take any four consecutive numbers in the sequence
 Multiply the middle two terms and double result
 Multiply the two outer numbers
 Add the squares of the inner two numbers
 The last three results form a Pythagorean triple
Curiosity 3
 The area of the previously generated Pythagorean
triangle is the product of the picked four consecutive
numbers
Definitions
Recall that a Pythagorean triple 𝑎, 𝑏, 𝑐 generates a
Pythagorean triangle with legs 𝑎 and 𝑏 and hypotenuse 𝑐
(according to our definition). Now given a Pythagorean
triple 𝑎, 𝑏, 𝑐 , we define its area 𝐴 and perimeter 𝑃 by
𝐴 = 𝑎𝑏/2,
𝑃 = 𝑎 + 𝑏 + 𝑐.
Notice that these agree with the geometrical areas and
perimeters of the corresponding Pythagorean triangle.
Curiosity 4
 The perimeters and areas of the Pythagorean triangles
have a fascinating connection with the Pythagorean
triples from which these triangles arise
 The only right triangle that satisfies 𝐴 ≤ 𝑃 is the 3-4-5
triangle
 The only two right triangles that satisfy 𝐴 = 𝑃 are the
6-8-10 and 5-12-13 triangles
 The only three right triangles that satisfy 𝐴 = 2𝑃 are
the 12-16-20, the 10-24-26, and the 9-40-41 triangles
Curiosity 5
 For every positive integer 𝑛, there exists a Pythagorean
triple where whose area is 𝑛 times its perimeter
 For every positive integer 𝑛, there exists a Pythagorean
triangle whose area is 𝑛 times its perimeter
Curiosity 6
 Unsolved problem
 It is not known whether there are two Pythagorean
triples 𝑎, 𝑏, 𝑐 and 𝑎′ , 𝑏 ′ , 𝑐′ such that 𝑎𝑏𝑐 = 𝑎′ 𝑏 ′ 𝑐′
Curiosity 7
 In 1643, Pierre Fermat proposed a challenge that he




answered himself (obviously)
Find 𝑎, 𝑏, 𝑐 ∈ 𝑆 such that 𝑎 + 𝑏 and 𝑐 are both perfect
squares
He came out with
4,565,486,027,761; 1,061,652,293,520; 4,687,298,610,289
Indeed, 𝑎 + 𝑏 = 2,372,1592 and 𝑐 = 2,165,0172
He also proved that this is the smallest triangle having this
property
Curiosity 8
 Generalized Pythagorean Theorem
 Are there triples (𝑎, 𝑏, 𝑐) with 𝑎𝑛 + 𝑏 𝑛 = 𝑐 𝑛 where 𝑛 ≥




3?
The answer is no!
Fermat claimed he had the proof and challenged
mathematicians for the next 357 years
This theorem became Fermat’s Last Theorem
…until 1994, when Andrew Wiles (1953–) prove it
References
 Dudley, Underwood. Elementary Number Theory.
Second Edition
 Katz, Victor J. A History of Mathematics An
Introduction. Third Edition
 Loomis, Elisha S. The Pythagorean Proposition
 Posamentier, Alfred S. The Pythagorean Theorem