A Note on Gaussian Integers, Pythagorean Triples, and Extended Pythagorean Triples
David Alger
The Citadel
Math 495
Section I – Introduction:
Pythagoras is credited for coming to the conclusion that the areas of the two squares
erected on the two legs of a right triangle would equal the area of the square erected on the
hypotenuse. This is known as the Pythagorean Theorem. Pythagoras, who had a mystical view
of integers [2], is credited with uncovering certain integer patterns which satisfied the “algebra”
version of the Pythagorean Theorem, that is integers a, b, and c with a2 + b2 = c2. We are
looking in this paper at these integers and also at the Gaussian Integers, that is, the set of
numbers of the form a + bi in which a and b are Integers and i   1 . Gaussian Integers can
also be described as the set of algebraic integers in the finite extension of field Q(i). The study
is to explore the connection between Pythagorean triples and Gaussian integers.
It was known to the Greeks and proved by Euler that if a number c could be factored as
c = c1 * c2 where each of c1 and c2 could be written as a sum of two squares, then c could be
written as a sum of two squares in two ways. Here, c1 = x2 + y2 and c2 = a2 + b2 and c = (x2 +
y2)*(a2 + b2).
We say (a, b, c) is a PT to mean it is a Pythagorean Triple, and if further, gcd (a,b,c) = 1,
then we say (a,b,c) is a Primitive Pythagorean Triple (or PPT)
Theorem 1 [Euclid’s 47th Proposition]: If (a, b, c) is a PPT, then a = 2mn, b = m2 – n2 and
c = m2 + n2 for some pair of integers m, n with
1. m > n
2. gcd(m, n) = 1
3. m, n = not both odd.
Many great mathematicians have had their share of successes with the Pythagorean
triples. They each came up with their own rule for finding numbers that make up Pythagorean
Triples. We list some of them below [3]:
1.
Rule of Pythagoras: Let n be odd; then the triple (n,
2
2
n2 1 n2 1
,
) is a PPT. Notice
2
2
 n2 1
4n 2  n 4  2n 2  1  n 2  1 
 
 . See Table 1.
n  
 
4
 2 
 2 
2
Plato’s Rule: Let m be any even number divisible by 4; then (m,
2.
2
m2
m2
 1 ) is a
1 ,
4
4
2
 m2

 m2

m4 m2
 1  m 2 

 1  
 1 . See Table 2.
PPT since m  
16
2
 4

 4

3.
Euclid’s Rule: Let x and y be any two even or odd numbers, such that x and y contain no
x  y  , and x  y  are three
common factor greater than 2, and xy is a square. Then xy ,
2
2
2
 
x 2  2 xy  y 2  x  y 
x y

such numbers. For xy  
  xy 
 .
4
 2 
 2 
m2  n2
4.
Rule of Maseres: Let m and n be any two even or odd, m > n, and
a square
2n
2
2
m2  n2
2 m n
integer. Then m ,
, and
are three such numbers. For
2n
2n
2
2
2
2
m 2  n 2 4m 2 n 2  m 4  2m 2  n 2  n 4  m 2  n 2 
 .
m 

 
2n
4n 2
 2n  
5.
Dickerson’s Rule: Let m and n be any two prime integers, one even and the other odd,
m>n and 2mn a square. Then m  2mn , n  2mn , and m  n  2mn is a PT. For
2
m 
 

2

2

2
2mn  n  2mn  m 2  n 2  2mn  2m 2mn  2n 2mn  m  n  2mn .
6.
If the two legs are represented by a and b and the hypotenuse is represented by c the
following relations apply when p and q are two integers and k = gcd (a, b, c):
a  k p2  q2
b  k 2 pq 




c  k p2  q2
When k is equal to one the triple (a, b, c) is said to be primitive.
Section II –Gaussian Integers and PTs:
There is a close connection with the set of Gaussian integers and the set of PT’s. In fact,
manipulating the Gaussian Integers you are able to get Pythagorean Triples. If you square any
Gaussian Integer you will get a new Gaussian Integer. The magnitudes of those coefficients are
the lengths of the legs of a right triangle. The hypotenuse is found by taking the norm of the
Gaussian Integer. The norm of a Gaussian Integer x  yi is x 2  y 2 . For example, using a
randomly selected z  3  2i , we see z 2  5  12i , and the length of z squared is 13. But then
(5,12,13) is a PT. We define, for any complex number z  a  bi , Re( z )  a , Im( z )  b . Here
“Re” refers to “real” and “Im” refers to “imaginary”. L(z) denotes the length z, namely
Norm(z) .
Alger 2
Theorem 2: If z  a  bi is any Gaussian Integer, then ( a 2  b 2 , 2ab , a 2  b 2 ) is a PT.
Proof 2:
The proof is immediate applying Theorem 1. We can restate the theorem as: if
z  a  bi is any Gaussian Integer, then ( Re( z 2 ) , Im( z 2 ) , L(z ) 2 ) is a PT. If
gcd a, b  1 , then the PT is a PPT.
Section III – Primes in the ring of Gaussian Integers
In addition to being defined slightly differently than normal integers, Gaussian integers
also have their own set of prime numbers. A Gaussian Prime can be defined as a number which
can not be factored as a product of two Gaussian Integers except by using a unit {±1, ±i}. For
example, since 13  32  2 2  3  2i 3  2i  we see 13 is not a Gaussian Prime even though it
is a prime in Z.
Definition: An element α in G = {a + bi : a,b  Z}is prime iff α is not a unit and whenever
α = xy (for x,y  G) either x or y is a unit.
The factorization for 13 is an example of a serious difficulty. Note that a 2  b 2 can
always be factored in G as a 2  b 2  a  bi a  bi  . This shows two is not a prime either since
2  12  12  1  i 1  i  . Since neither 1  i nor 1  i are units, 2 (by the definition) can not be
a Gaussian prime.
Theorem 3: No rational prime p is a Gaussian prime if p  a 2  b 2 for some rational
integers a, b.
Proof 3:
The difficult question is this: which rational primes can be written as a sum of
two squares. One may also ask: what rational integers can be written as a sum of two
squares?
2n 2  2m2  4n 2  4m 2  4n 2  m 2  , a multiple of 4.
2n 2  2m  12  4n 2  4m 2  4m  1  4k  1  1mod 4
2n  12  2m  12  4n 2  4n  1  4m 2  4m  1  4 j  2
These show that a 2  b 2 is 0 mod 4 , 1mod 4 , or 2 mod 4 . But a 2  b 2  3mod 4 .
The point for this paper is that a rational prime q which satisfies q  3mod 4 might be
a Gaussian prime – but no other rational primes can possibly be Gaussian primes.
Numbers that are rational prime integers are not necessarily Gaussian prime integers and
vice versa. The Gaussian primes are those rational primes that can not be expressed by the sum
of two squares.
In summary, the definition is saying is that a number a + bi (b ≠ 0) is prime in G if its
norm is prime in Z. N(xy) = N(x)N(y) = prime (then N(x) or N(y) = 1) and so x or y is a unit.
The following are two simple characteristics that would classify a number as a prime in G:
1.
Primes in Z which are 3 mod 4.
2.
x = a + bi for N(x) = prime.
Alger 3
Section IV - Extending the Pythagorean Triples:
When taking multiple triples together an extended version of a Pythagorean equation
may be made. For example: 3 2  4 2  5 2 and 5 2  12 2  13 2 therefore 32  4 2  12 2  132 .
This can be broken down into the following general equations: a 2  b 2  c 2 and c 2  d 2  e 2
therefore a 2  b 2  d 2  e 2 .
a
3
5
7
9
Extended Pythagorean Triples
b
c
d
4
5
12
12
13
84
24
25
312
40
41
840
e
13
85
313
841
This property can be used when putting two right triangles adjacent to each other in
order to find the side lengths for larger polygons.
Two examples 12  2 2  2 2  3 2 and 2 2  3 2  6 2  7 2 suggested a pattern which led to
the infinite set of EPTs in Table 3 of the Appendix.
Theorem 4: Suppose (n, n+1, n(n+1), n(n+1)+1) is an EPT
Proof 4:
If it is an EPT then n2 + (n + 1)2 + (n(n + 1))2 = (n(n + 1) + 1)2
n2 + (n + 1)2 + (n2 + n) = (n2 + n + 1)2
n2 + n2 + 2n + 1 + n4 + 2n3 + n2 = n4 + 2n3 + 3n2 + 2n + 1
Since this is true the set of numbers (n, n+1, n(n+1), n(n+1)+1) is an EPT.
It is possible to turn many familiar algebra identities into new sources of EPTs. We
2
illustrate the idea with 1  2n  n 2  n  1 in the next theorem.
Theorem 5: Suppose 2n is a square. Then (1, 2n , n, n  1 ) is an EPT.
Proof 5:
In order for (1, 2n , n, n  1 ) to be an EPT the following relationship must
exist.
12 + 2n 2+ n2 = ( n  1 )2
Since 12 + 2n 2+ n2 = 12 + 2n+ n2 = ( n  1 )2 the following relationship is an EPT.
In order to construct more “formula based EPTs the only thing really necessary is to
construct an equality with one side of the equations equaling a three term polynomial (ie
3 2  4 2  12 2  13 2 ) and the square root of each term equaling an integer. For example, taking
the equality a 2  b 2  c 2  x 2 and manipulating one or two of the variables you are able to
make more generalized equations. By setting a equal to a constant and x equal to a modification
Alger 4
of c you could get a whole set of answers. By varying the constant you will get different sets of
answers. The following theorem will demonstrate one of those sets.
Theorem 6: Suppose the equation a 2  b 2  c 2  x 2 is true. Set a equal to a constant and x
equal to c + a.
Proof 6:
By substitution you see that 12  b 2  c 2  (c  1) 2 . So 1  b 2  c 2  c 2  2c  1
and the c2 terms and the ones cancel out to get b 2  2c  1 . By simplifying and solving
b2
for c you get it to be equal to
. After putting that in a table you realize that as long as
2
a is equal to one and b is an even number you will always get an EPT. See Table 4.
By generalizing that theorem even more and testing for different values of the constant
we come up with an Über theorem as follows:
Über Theorem: Suppose the equation a 2  b 2  c 2  x 2 is true. By setting a equal to a constant
b2
and x set equal to c + a, then c will be equal to
. If a is an even number then a EPT
2a
will be formed for every value of b equal to a*n. If a is an odd number then a EPT will
be formed for every value of b equal to 2*a* n. Where n  Z.
Über Proof: By substitution the equation becomes a 2  b 2  c 2  (c  a) 2  c 2  2ac  a 2 . By
 b2 
simplifying we get b 2  2ac and by further substitution b 2  2a  we show that all
 2a 
terms cancel out so the equation is in fact equivalent. See Table 5 for examples of
various constant values.
Alger 5
Appendix:
n
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Table 1
(n -1)/2
(n2+1)/2
4
5
12
13
24
25
40
41
60
61
84
85
112
113
144
145
180
181
220
221
264
265
312
313
364
365
420
421
480
481
544
545
612
613
684
685
760
761
840
841
2
n+1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Table 3
n(n+1)
2
6
12
20
30
42
56
72
90
110
132
156
182
210
240
272
306
342
380
420
m
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
n(n+1)+1
3
7
13
21
31
43
57
73
91
111
133
157
183
211
241
273
307
343
381
421
a
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Alger 6
Table 2
m2/4-1 m2/4+1
3
5
15
17
35
37
63
65
99
101
143
145
195
197
255
257
323
325
399
401
483
485
575
577
675
677
783
785
899
901
1023
1025
1155
1157
1295
1297
1443
1445
1599
1601
b
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
Table 4
c = b2/2
2
8
18
32
50
72
98
128
162
200
242
288
338
392
450
512
578
648
722
800
x=c+1
3
9
19
33
51
73
99
129
163
201
243
289
339
393
451
513
579
649
723
801
n
a
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
Table 5
b
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
6
12
18
24
30
36
42
48
54
60
4
8
12
16
20
24
28
32
36
40
10
20
30
40
50
60
70
80
90
100
c
x
2
8
18
32
50
72
98
128
162
200
1
4
9
16
25
36
49
64
81
100
6
24
54
96
150
216
294
384
486
600
2
8
18
32
50
72
98
128
162
200
10
40
90
160
250
360
490
640
810
1000
3
9
19
33
51
73
99
129
163
201
3
6
11
18
27
38
51
66
83
102
9
27
57
99
153
219
297
387
489
603
6
12
22
36
54
76
102
132
166
204
15
45
95
165
255
365
495
645
815
1005
Alger 7
Works Cited:
[1]
A. H. Beiler, Recreations in the Theory of Numbers, Second Edition, Dover Publications,
INC., New York, 1966.
[2]
D. M. Burton, The History of Mathematics, Allyn and Bacon, INC., Boston,
Massachusetts, 1985.
[3]
E. S. Loomis, The Pythagorean Proposition, Second Edition, National Council of
Teachers of Mathematics, Ann Arbor, Michigan, 1940.
Alger 8