Pythagorean triples and its
extensions
Maths IA
Mingke Peng
Contents
1. Introduction ........................................................................................................................................ 2
2. Primitive Pythagorean Triples ............................................................................................................. 2
3. Properties of Pythagorean Triples ...................................................................................................... 3
4. Pythagorean Quadruples .................................................................................................................... 5
6. Conclusion ........................................................................................................................................... 7
Bibliography ............................................................................................................................................ 8
1. Introduction
The opportunity for exploration and the wide-ranging applications of Pythagoras’ theorem is what
initially sparked my interest in the topic. Pythagoras’ theorem provides a foundation for much of
Euclidean geometry, and being equivalent to Euclid’s 5th Postulate, it is almost axiomatic. However,
its significance is not limited to geometry – as the theorem can be stated algebraically:
𝑎2 + 𝑏 2 = 𝑐 2
When a, b, c are integers, the solutions to the Diophantine equation are Pythagorean Triples, and
these have interesting properties in the context of number theory. I was able to explore an aspect
while reading the C J Bradley’s Introduction to Number Theory, but the book got me thinking about
the properties of Pythagorean Triples in terms of modular arithmetic, as well as how the idea of
Pythagorean Triples could be extended. My aim in this investigation will be to prove some of these
properties and demonstrate how they extend to Pythagorean Quadruples, to determine how closely
linked they are. Pythagorean Triples which are integers a, b, c, d which satisfy
𝑎2 + 𝑏 2 + 𝑐 2 = 𝑑 2 .
2. Primitive Pythagorean Triples
In this investigation, only primitive Pythagorean Triples – ones where all the numbers in the triplet
are coprime – will be studied, as all others can be trivially formed by multiplying each number in a
primitive triple by an integer.
All triples formed in this way still satisfy Pythagoras’ Theorem.
Proof (1):
Suppose a, b, c are integers such that 𝑎2 + 𝑏 2 = 𝑐 2 .
Multiplying each of a, b, c by k ∈ ℤ+ , we get ka, kb, kc.
(𝑘𝑎)2 + (𝑘𝑏)2 = 𝑘 2 𝑎2 + 𝑘 2 𝑏 2
= 𝑘 2 (𝑎2 + 𝑏 2 )
Since 𝑎2 + 𝑏 2 = 𝑐 2 ,
𝑘 2 (𝑎2 + 𝑏 2 ) = 𝑘 2 (𝑐 2 )
= (𝑘𝑐)2
∴ (𝑘𝑎)2 + (𝑘𝑏)2 = (𝑘𝑐)2
Q.E.D.
3. Properties of Pythagorean Triples
We can start by listing some Pythagorean Triples, and then look for patterns within the numbers.
a
3
5
7
8
9
11
12
13
15
16
17
b
4
12
24
15
40
60
35
84
112
63
144
c
5
13
25
17
41
61
37
85
113
65
1451
From the table, an initial observation can be made that a is either odd or a multiple of four. It can
therefore be conjectured that:
a is never 2 mod 4. (Property 1)
Proof (2):
𝑎2 + 𝑏 2 = 𝑐 2
𝑐 2 − 𝑏 2 = 𝑎2
(𝑐 + 𝑏)(c − 𝑏) = 𝑎2
Now the difference between 𝑐 + 𝑏 and 𝑐 − 𝑏 is 2𝑏, so if 𝑐 + 𝑏 is odd then so is 𝑐 − 𝑏, and hence a
would be odd.
Similarly, if 𝑐 + 𝑏 is even, then so is a.
Squares are 0 mod 4 if even and 1 mod 4 if odd2, so 𝑐 2 − 𝑏 2 is 0 or 1 mod 4, which means there are
3 possibilities:
(1) 𝑐 2 ≡ 𝑏 2 ≡ 1 (𝑚𝑜𝑑 4) ⇒ 𝑎2 ≡ 0 (𝑚𝑜𝑑 4)
(2) 𝑐 2 ≡ 𝑏 2 ≡ 0 (𝑚𝑜𝑑 4) ⇒ 𝑎2 ≡ 0 (𝑚𝑜𝑑 4)
(3) 𝑐 2 ≡ 1 (𝑚𝑜𝑑 4), 𝑏 2 ≡ 0 (𝑚𝑜𝑑 4) ⇒ 𝑎2 ≡ 1 (𝑚𝑜𝑑 4)
However, in (2), a, b, c are all multiples of 4, which means they have a common factor of 4. This
means they cannot be primitive triples.
1
Chandrahas Halai. Triples and quadruples: from Pythagoras to Fermat https://plus.maths.org/content/triplesand-quadruples Accessed 23/6/2018
2
https://proofwiki.org/wiki/Square_Modulo_4 Accessed 23/6/2018
Therefore, only possibilities (1) and (3) remain.
In possibility (3), a is odd as 𝑎2 ≡ 1 (𝑚𝑜𝑑 4), i.e. a ≡ 1 𝑜𝑟 3 (𝑚𝑜𝑑 4), so it does not contradict the
conjecture.
Suppose 𝑎 ≡ 2 (𝑚𝑜𝑑 4).
𝑐 2 − 𝑎2 = 𝑏 2
(𝑐 + 𝑎)(c − 𝑎) = 𝑏 2
c ≡ 1 𝑜𝑟 3 (𝑚𝑜𝑑 4),
∴ 𝑐 + 𝑎 ≡ 1 𝑜𝑟 3 (𝑚𝑜𝑑 4) and 𝑐 − 𝑎 ≡ 3 𝑜𝑟 1 (𝑚𝑜𝑑 4) respectively.
∴ (𝑐 + 𝑎)(c − 𝑎) = 𝑏 2 ≡ 3 (𝑚𝑜𝑑 4)
However, this contradicts the previous assertion that 𝑏 2 ≡ 1 (𝑚𝑜𝑑 4), so the supposition that
𝑎 ≡ 2 (𝑚𝑜𝑑 4) must be false.
Q.E.D.
An identical proof can be used to show that b is never 2 mod 4.
In proving property 1, we have also proved that:
c is always odd, as 𝑐 2 ≡ 1 (𝑚𝑜𝑑 4). (Property 2)
One of 𝑎2 and 𝑏 2 is 0 (mod 4) and the other is 1 (mod 4) ⇒ exactly one of a and b is odd (Property 3)
It can also be proved that c itself is congruent to 1 mod 4.
Another observation to be made is that exactly one of a and b is 0 mod 3 or mod 4, while exactly one
of a, b, c is 0 mod 5, and further, these observations can be shown to be true3. It can also be noted
that the product of the smallest triple is 60.
With further investigation, it seems the products of all the triples are multiples of 60. (Property 4)
Proof:
If we prime factorise the product abc, then 3, 4 and 5 will be prime factors, as mentioned above.
This means that 3 × 4 × 5 = 60 must be a factor of abc.
Q.E.D.
Furthermore, 60 must be the largest number which divides the product of any given Pythagorean
Triple, as the product of the smallest Pythagorean Triple is 60, and a divisor of any number cannot
be larger than the number itself.
3
Waclaw Sierpinski. Pythagorean Triangles.
https://books.google.co.uk/books?id=6vOfpjmCd7sC&pg=PR4&redir_esc=y#v=onepage&q&f=false Accessed
1/7/2018.
4. Pythagorean Quadruples
Perhaps unexpectedly, Pythagorean Quadruples also have geometrical significance. They can be
viewed as dimensions of a cuboid, where a, b, c are the lengths of the sides and d is the length of the
diagonal.
4
Similarly to Proof (1), it can be shown that for a, b, c, d which form a Pythagorean quadruple, ka, kb,
kc, kd also form a Pythagorean quadruple for all k ∈ ℤ+ . This property is hence conserved as the
number of dimensions involved increases from 2 to 3.
It will be interesting to discover whether or not other properties of Pythagorean Triples shown in
chapter 3 also apply to Pythagorean Quadruples.
To verify if this can reasonably be the case, we can again list some of the smallest primitive
Pythagorean Quadruples.
a
1
1
1
1
1
2
2
2
2
3
3
3
4
4
4
4
b
2
4
6
8
18
3
5
6
10
4
6
8
5
7
8
c
2
8
18
32
30
6
14
9
11
12
22
36
20
32
19
d
3
9
19
33
35
7
15
11
15
13
23
37
21
33
21
Chandrahas Halai. Triples and quadruples: from Pythagoras to Fermat https://plus.maths.org/content/triplesand-quadruples Accessed 25/6/2018
We can immediately see that the first listed Pythagorean Quadruple disproves Property 1 for
quadruples, as it is a counterexample.
However, property 2 seems to hold for Pythagorean Quadruples (meaning that d is always odd), as
there are no counterexamples in the list.
Proof:
𝑎2 , 𝑏 2 , 𝑐 2 , 𝑑2 are all 0 𝑜𝑟 1 (𝑚𝑜𝑑 4)
max(𝑎2 + 𝑏 2 + 𝑐 2 (𝑚𝑜𝑑 4)) = 3
This means there are 4 distinct configurations:
(1) 𝑎2 = 0 (𝑚𝑜𝑑 4), 𝑏 2 = 0 (𝑚𝑜𝑑 4), 𝑐 2 = 0 (𝑚𝑜𝑑 4) ⇒ 𝑑2 = 0 (𝑚𝑜𝑑 4)
(2) 𝑎2 = 1 (𝑚𝑜𝑑 4), 𝑏 2 = 0 (𝑚𝑜𝑑 4), 𝑐 2 = 0 (𝑚𝑜𝑑 4) ⇒ 𝑑2 = 1 (𝑚𝑜𝑑 4)
(3) 𝑎2 = 0 (𝑚𝑜𝑑 4), 𝑏 2 = 1 (𝑚𝑜𝑑 4), 𝑐 2 = 0 (𝑚𝑜𝑑 4) ⇒ 𝑑2 = 1 (𝑚𝑜𝑑 4)
(4) 𝑎2 = 0 (𝑚𝑜𝑑 4), 𝑏 2 = 0 (𝑚𝑜𝑑 4), 𝑐 2 = 1 (𝑚𝑜𝑑 4) ⇒ 𝑑2 = 1 (𝑚𝑜𝑑 4)
(2), (3) and (4) are effectively the same, as a, b and c are interchangeable.
Only (1) needs to be eliminated, as the others do not contradict the statement.
If all of a, b, c, d are 0 (mod 4), then they are all multiples of 4. This means that they have a common
factor of 4, which means they cannot be primitive. Therefore, (1) is impossible when only
considering primitive quadruples.
∴ 𝑑2 = 1 (𝑚𝑜𝑑 4)
∴ 𝑑 is odd.
Q.E.D.
There are also suggestions from the list of quadruples that Property 3 holds. In fact, this follows from
the proof above, as
𝑎2 = 1 (𝑚𝑜𝑑 4), 𝑏 2 = 0 (𝑚𝑜𝑑 4), 𝑐 2 = 0 (𝑚𝑜𝑑 4) ⇒ 1 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑢𝑡 𝑜𝑓 𝑎, 𝑏, 𝑐.
It is evident that the precise definition of Property 4 does not hold true for Pythagorean Quadruples,
as 60 > 1 × 2 × 2 × 3 = 12. However, is 12 a factor of the products of all Pythagorean Quadruples?
Proof:
There are at least two even numbers in a, b, c, d, which means that 4 is a factor of abcd.
Therefore, to prove the result, it needs to be proved that at least one of abcd is a multiple of 3.
5. Conclusion
The Diophantine equations which define Pythagorean Triples and Quadruples may look very similar,
but
Bibliography
https://plus.maths.org/content/triples-and-quadruples (Accessed 22/6/2018)
https://proofwiki.org/wiki/Square_Modulo_4 (Accessed 24/6/2018)