Joel Malissa
My Pythagorean Triples Research
a2 + b2 = c2: The Pythagorean theorem was discovered roughly 2,500 years ago, and its integer
solutions, Pythagorean triples, date all the way back to stone structures built in 2500 BC. A simple
curiosity quickly developed into one goal: three formulas (one each for a, b and c) that generate all
primitive Pythagorean triples. From a list of primitive triples, the differences between c and b appear
limited to a rather small list, starting 1, 2, 8, 9… The famous Greek philosopher Plato, used these
formulas for finding triples:
a = 2n
b = n2 – 1
c = n2 + 1
I wrote programs in TI-BASIC to find all possible c - b values less than 1,000. This sequence,
which I refer to as the Pythagorean differences, yields many intriguing patterns. Starting with the
individual differences, the most obvious pattern is that all odd perfect squares are Pythagorean
differences. Furthermore, two raised to any odd integer exponent also generates a Pythagorean
difference. In fact, all other Pythagorean differences, other than the case in which c - b = 1, are products
of these two sets of numbers. Consequently, all Pythagorean differences can be represented by
where m is either zero or a positive odd integer, and n is an odd integer. This
statement is my first significant, and certainly most elegant, finding from all my research.
Looking at all the Pythagorean differences collectively also yields notable patterns. To start,
graphing these differences reveals a very strong quadratic trend:
1000
800
y = 0.6813x2 + 0.1589x - 0.7541
R² = 0.9991
c-b
600
400
200
0
Clearly, Pythagorean differences follow a parabolic curve.
Looking back at the data, there seemed to be a pattern in the changing of the c - b values, so I
graphed that too:
90
80
70
Δ(c - b)
60
50
40
30
20
10
0
There seems to be some sort of predictability to the fluctuating nature of this graph, so I generated a
Fourier transform for the first 32 values of Δ(c - b). (Only the first half of the symmetrical graph is
shown.)
Frequency (Hz)
Fourier transform of Δ(c - b)
800
700
600
500
400
300
200
100
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17
The evident spike at 14 Hz indicates that the Δ(c - b) values can be modeled to an extent by a sine wave
with a period of repetition of 32/14, and the pronounced height of this spike relative to its neighboring
points highly suggests, almost conclusively, that there must exist a well-defined pattern in the plot of
Δ(c - b) and by extension, in Pythagorean differences.
Extending beyond the Pythagorean differences themselves, infinitely many primitive
Pythagorean triples occur for each Pythagorean difference. (I wrote another program, this time using
Java, to formulate and test all of the equations and triples presented here.) The simplest set of triples
occurs with c – b = 1:
a1 = 2n + 1
b1 = 2n2 + 2n
c1 = 2n2 + 2n + 1
Throughout, n will always represent any positive integer in these “equation trios”. For this instance, the
following primitive triples are generated, starting with n = 1:
(3, 4, 5)
(5, 12, 13)
(7, 24, 25)
(9, 40, 41)
…
(As a side note, all of these triples form the large majority of a special category known as twin
Pythagorean triples, triples involving at least two consecutive integers.) Notice that a utilizes a linear
equation, while b and c are quadratic. All Pythagorean differences share this simplicity in generating
triples, but not necessarily primitive ones.
Consider the instance c – b = 9. The following equations create an infinite number of triples:
a9 = 6n + 27
b9 = 2n2 + 18n + 36
c9 = 2n2 + 18n + 45
However, setting n equal to 3 generates the triple (45, 108, 117), which can be simplified by dividing by
9 to achieve (5, 12, 13). Since the goal is to discover all primitive triples, I searched for more exclusive
solutions. Two crucial findings (that I realized on a school bus) led me to generating only the desired
triples. I like to call them mathematical logical operators:
MATHEMATICAL LOGICAL OPERATORS
Even
Odd
n
(1-(-1)n)/2
(1+(-1) )/2
input (n)
output input (n)
output
even
1 even
0
odd
0 odd
1
Using the odd operator, an expression arose for a9 that resembled 9n + {3, 0} + 21, where “+ {3, 0}”
meant add three if n is odd, else add zero. After filling in all the analogous cases, I moved on to
calculating b9. Using a similar technique, two intertwined quadratics, one for odd n values and the other
for even, uniquely accounted for the desired outcomes. Lastly, c was the easiest to determine, for the
difference between b and c is set by the Pythagorean difference. Finally, I am able to equate endless
primitive triples to Pythagorean differences that generate either constantly or alternating primitive
triples.
In the future, I hope to discover additional and more efficient methods to seek primitive
Pythagorean triples. Eventually, my goal is to be able to generate all of the solutions to every
Pythagorean difference, and find hidden patterns within these answers. With a little luck and much
work, these trends will hopefully yield general conclusions concerning all primitive Pythagorean triples
both collectively and elegantly.
Prime
First Primitive
c - b Factorization
Triple
1
(1)
3
4
5
2
2
8 15 17
nth Primitive Triple
a
b
2n+1
4n+4
c
2 n^2 + 2 n
4 n^2 + 8 n + 3
2 n^2 + 2 n + 1
4 n^2 + 8 n + 5
8
23
20
21
29
8 n + 12
4 n^2 + 12 n + 5
4 n^2 + 12 n + 13
9
32
33
56
65
(3/2)(15 -(-1)^n +6 n)
(1/4) (95 - 15 (-1)^n - 6 (-15 + (-1)^n) n + 18 n^2)
(1/4) (131 - 15 (-1)^n - 6 (-15 + (-1)^n) n + 18 n^2)
18
2*32
48
55
73
3 (9 - (-1)^n + 6 n)
(1/2) (23 - 9 (-1)^n) - 3 (-9 + (-1)^n) n + 9 n^2
(1/2) (59 - 9 (-1)^n) - 3 (-9 + (-1)^n) n + 9 n^2
25
52
65
72
97
32
25
88 105 137
16 n + 72
4 n^2 + 36 n + 65
4 n^2 + 36 n + 97
49
72 119 120 169
50
2*52 140 171 221
72
23*32 204 253 325
6 (27 - (-1)^n + 6 n)
293/2 - 27 (-1)^n/2 + 81 n - 3 (-1)^n n + 9 n^2
437/2 - 27 (-1)^n/2 + 81 n - 3 (-1)^n n + 9 n^2
81
34 207 224 305
(9/2) (39 - (-1)^n + 6 n)
(1/4) (599 - 39 (-1)^n - 6 (-39 + (-1)^n) n + 18 n^2)
(1/4) (923 - 39 (-1)^n - 6 (-39 + (-1)^n) n + 18 n^2)
98
2*72 252 275 373
121
112 297 304 425
128
27 336 377 505
32 n + 304
4 n^2 + 76 n + 297
4 n^2 + 76 n + 425
162
2*34 396 403 565
9 (39 + (-1)^n + 6 n)
(1/2) (599 + 39 (-1)^n) + 3 (39 + (-1)^n) n + 9 n^2
(13/2) (71 + 3 (-1)^n) + 3 (39 + (-1)^n) n + 9 n^2
12 (51 - (-1)^n + 6 n)
(1/2) (1013 - 51 (-1)^n) - 3 (-51 + (-1)^n) n + 9 n^2
(1/2) (1589 - 51 (-1)^n) - 3 (-51 + (-1)^n) n + 9 n^2
64 n + 1184
4 n^2 + 148 n + 1113
4 n^2 + 148 n + 1625
18 (87 - (-1)^n + 6 n)
(1/2) (3137 - 87 (-1)^n) - 3 (-87 + (-1)^n) n + 9 n^2
(1/2) (4433 - 87 (-1)^n) - 3 (-87 + (-1)^n) n + 9 n^2
36 1809 1880 2609 (27/2) (129 + (-1)^n + 6 n)
(1/4) (6863 + 129 (-1)^n + 6 (129 + (-1)^n) n + 18 n^2)
(1/4) (9779 + 129 (-1)^n + 6 (129 + (-1)^n) n + 18 n^2)
169
200
132 429 460 629
3
2 *52 540 629 829
225
32*52 555 572 797
242
2*112 616 663 905
288
25*32 696 697 985
289
172 731 780 1069
338
2*132 832 855 1193
361
392
192 893 924 1285
3
2 *72 1036 1173 1565
441
32*72 1113 1184 1625
450
2*32*52 1140 1219 1669
512
29 1248 1265 1777
529
232 1311 1360 1889
578
2*172 1428 1475 2053
54 1525 1548 2173
625
3
648
2 *34 1692 1885 2533
722
2*192 1748 1755 2477
729
800
841
882
5
2 *52 1960 2001 2801
292 2059 2100 2941
2
2*3 *72 2184 2263 3145
961
312 2325 2332 3293
968
23*112 2508 2765 3733