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 . 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
- m n
- gcd(m, n) = 1
- 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]:
- Rule of Pythagoras: Let n be odd; then the triple (n, , ) is a PPT. Notice . See Table 1.
- Plato’s Rule: Let m be any even number divisible by 4; then (m, , ) is a PPT since . See Table 2.
- Euclid’s Rule: Let x and y be any two even or odd numbers, such that x and y contain no common factor greater than 2, and xy is a square. Then , , and are three such numbers. For .
- Rule of Maseres: Let m and n be any two even or odd, m n, and a square integer. Then m2, , and are three such numbers. For .
- Dickerson’s Rule: Let m and n be any two prime integers, one even and the other odd, mn and 2mn a square. Then , , and is a PT. For .
- 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):
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 is . For example, using a randomly selected , we see , and the length of z squared is 13. But then (5,12,13) is a PT. We define, for any complex number , , . Here “Re” refers to “real” and “Im” refers to “imaginary”. L(z) denotes the length z, namely .
Theorem 2: If is any Gaussian Integer, then (,,) is a PT.
Proof 2: The proof is immediate applying Theorem 1. We can restate the theorem as: if is any Gaussian Integer, then (,,) is a PT. If , 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 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,bZ}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 can always be factored in G as . This shows two is not a prime either since . Since neither nor are units, 2 (by the definition) can not be a Gaussian prime.
Theorem 3: No rational prime p is a Gaussian prime if 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?
, a multiple of 4.
These show that is ,, or . But .
The point for this paper is that a rational prime q which satisfies 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.
Section IV - Extending the Pythagorean Triples:
When taking multiple triples together an extended version of a Pythagorean equation may be made. For example: and therefore . This can be broken down into the following general equations: and therefore .
Extended Pythagorean Triplesa / b / c / d / e
3 / 4 / 5 / 12 / 13
5 / 12 / 13 / 84 / 85
7 / 24 / 25 / 312 / 313
9 / 40 / 41 / 840 / 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 and 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 illustrate the idea with in the next theorem.
Theorem 5: Suppose 2n is a square. Then (1, , n, ) is an EPT.
Proof 5: In order for (1, , n, ) to be an EPT the following relationship must exist.
12 + 2+ n2 = ()2
Since 12 + 2+ n2 = 12 + 2n+ n2 = ()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 ) and the square root of each term equaling an integer. For example, taking the equality 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 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 is true. Set a equal to a constant and x equal to c + a.
Proof 6: By substitution you see that . So and the c2 terms and the ones cancel out to get . By simplifying and solving for c you get it to be equal to . After putting that in a table you realize that as long as 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 is true. By setting a equal to a constant and x set equal to c + a, then c will be equal to . If a is an even number then a EPT 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 . By simplifying we get and by further substitution we show that all terms cancel out so the equation is in fact equivalent. See Table 5 for examples of various constant values.
Appendix:
n / (n2-1)/2 / (n2+1)/2
3 / 4 / 5
5 / 12 / 13
7 / 24 / 25
9 / 40 / 41
11 / 60 / 61
13 / 84 / 85
15 / 112 / 113
17 / 144 / 145
19 / 180 / 181
21 / 220 / 221
23 / 264 / 265
25 / 312 / 313
27 / 364 / 365
29 / 420 / 421
31 / 480 / 481
33 / 544 / 545
35 / 612 / 613
37 / 684 / 685
39 / 760 / 761
41 / 840 / 841
Table 2
m / m2/4-1 / m2/4+1
4 / 3 / 5
8 / 15 / 17
12 / 35 / 37
16 / 63 / 65
20 / 99 / 101
24 / 143 / 145
28 / 195 / 197
32 / 255 / 257
36 / 323 / 325
40 / 399 / 401
44 / 483 / 485
48 / 575 / 577
52 / 675 / 677
56 / 783 / 785
60 / 899 / 901
64 / 1023 / 1025
68 / 1155 / 1157
72 / 1295 / 1297
76 / 1443 / 1445
80 / 1599 / 1601
Table 3
n / n+1 / n(n+1) / n(n+1)+1
1 / 2 / 2 / 3
2 / 3 / 6 / 7
3 / 4 / 12 / 13
4 / 5 / 20 / 21
5 / 6 / 30 / 31
6 / 7 / 42 / 43
7 / 8 / 56 / 57
8 / 9 / 72 / 73
9 / 10 / 90 / 91
10 / 11 / 110 / 111
11 / 12 / 132 / 133
12 / 13 / 156 / 157
13 / 14 / 182 / 183
14 / 15 / 210 / 211
15 / 16 / 240 / 241
16 / 17 / 272 / 273
17 / 18 / 306 / 307
18 / 19 / 342 / 343
19 / 20 / 380 / 381
20 / 21 / 420 / 421
Table 4
a / b / c = b2/2 / x = c + 1
1 / 2 / 2 / 3
1 / 4 / 8 / 9
1 / 6 / 18 / 19
1 / 8 / 32 / 33
1 / 10 / 50 / 51
1 / 12 / 72 / 73
1 / 14 / 98 / 99
1 / 16 / 128 / 129
1 / 18 / 162 / 163
1 / 20 / 200 / 201
1 / 22 / 242 / 243
1 / 24 / 288 / 289
1 / 26 / 338 / 339
1 / 28 / 392 / 393
1 / 30 / 450 / 451
1 / 32 / 512 / 513
1 / 34 / 578 / 579
1 / 36 / 648 / 649
1 / 38 / 722 / 723
1 / 40 / 800 / 801
Table 5
n / a / b / c / x
1 / 1 / 2 / 2 / 3
2 / 1 / 4 / 8 / 9
3 / 1 / 6 / 18 / 19
4 / 1 / 8 / 32 / 33
5 / 1 / 10 / 50 / 51
6 / 1 / 12 / 72 / 73
7 / 1 / 14 / 98 / 99
8 / 1 / 16 / 128 / 129
9 / 1 / 18 / 162 / 163
10 / 1 / 20 / 200 / 201
1 / 2 / 2 / 1 / 3
2 / 2 / 4 / 4 / 6
3 / 2 / 6 / 9 / 11
4 / 2 / 8 / 16 / 18
5 / 2 / 10 / 25 / 27
6 / 2 / 12 / 36 / 38
7 / 2 / 14 / 49 / 51
8 / 2 / 16 / 64 / 66
9 / 2 / 18 / 81 / 83
10 / 2 / 20 / 100 / 102
1 / 3 / 6 / 6 / 9
2 / 3 / 12 / 24 / 27
3 / 3 / 18 / 54 / 57
4 / 3 / 24 / 96 / 99
5 / 3 / 30 / 150 / 153
6 / 3 / 36 / 216 / 219
7 / 3 / 42 / 294 / 297
8 / 3 / 48 / 384 / 387
9 / 3 / 54 / 486 / 489
10 / 3 / 60 / 600 / 603
1 / 4 / 4 / 2 / 6
2 / 4 / 8 / 8 / 12
3 / 4 / 12 / 18 / 22
4 / 4 / 16 / 32 / 36
5 / 4 / 20 / 50 / 54
6 / 4 / 24 / 72 / 76
7 / 4 / 28 / 98 / 102
8 / 4 / 32 / 128 / 132
9 / 4 / 36 / 162 / 166
10 / 4 / 40 / 200 / 204
1 / 5 / 10 / 10 / 15
2 / 5 / 20 / 40 / 45
3 / 5 / 30 / 90 / 95
4 / 5 / 40 / 160 / 165
5 / 5 / 50 / 250 / 255
6 / 5 / 60 / 360 / 365
7 / 5 / 70 / 490 / 495
8 / 5 / 80 / 640 / 645
9 / 5 / 90 / 810 / 815
10 / 5 / 100 / 1000 / 1005
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 5