A Tale of Two Legs: Generating Pythagorean Triples from Odd and Even Beginnings

                                                                                                                           By Vishnu Vinjamuri

Introduction

This article explores a novel classification of Pythagorean triples based on whether the smallest leg is odd or even, leading to surprisingly elegant formulas and probabilistic insights.

The renowned Pythagorean theorem asserts that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). This relationship can be expressed with the formula a² + b² = c² , where 'c' denotes the hypotenuse, while 'a' and 'b' represent the two legs of the triangle.

Fig 1. Pictorial representation of Pythagoras theorem

Here we will explore the properties and characteristics of the narrowest possible right-angled triangles, where all three sides are measured in integer units.

Definition – Pythagorean Triple:

When a, b and c can be expressed as integers, they are called a Pythagorean triple, with c representing the length the hypotenuse.

In Fig 1, as the angle q becomes smaller, the hypotenuse c and the longer side b move closer each other.

Very narrow triangles can be constructed where c=b+1 or b+2, while still maintaining a as an integer.

In Plus magazine, Chandrahas Halai explored intriguing Pythagorean triples. For instances where the smallest number a is odd, the corresponding hypotenuse c exceeds b by just 1, resulting in the relationship c = b + 1. Conversely, when a is even and greater than 2, c exceeds b by 2, thus c = b + 2. It is evident that these configurations yield the narrowest possible right-angled triangles that can be formed using integer values.

Table 1 – Pythagorean Triples with the Smallest Hypotenuse from Odd Legs:

a	b	c
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
43	924	925
45	1012	1013
47	1104	1105
49	1200	1201
51	1300	1301
53	1404	1405
55	1512	1513
57	1624	1625
59	1740	1741

 

Table 2 – Pythagorean Triples with the Smallest Hypotenuse from Even Legs:

a	b	c
4	3	5
6	8	10
8	15	17
10	24	26
12	35	37
14	48	50
16	63	65
18	80	82
20	99	101
22	120	122
24	143	145
26	168	170
28	195	197
30	224	226
32	255	257
34	288	290
36	323	325
38	360	362
40	399	401
42	440	442
44	483	485
46	528	530
48	575	577
50	624	626
52	675	677
54	728	730
56	783	785
58	840	842
60	899	901

 

In Table 1, the three legs a,b,c are of the form  respectively.

In Table 2, a,b,c are of the form 2n, n²-1 and n²+1, which is straightforward.

Renaming Hypotenuses and Reframing the Formula:

From Table 1, we have the hypotenuse of least possible integer length, constructed from a side of least possible integer length in odd numbered units. Let us rename this hypotenuse as Minimal Odd Leg Hypotenuse or MOLH.

If we keenly observe, the MOLHs can be written in the form 2n-1, 2n*(n-1), and 2n*(n-1)+1, where 2n-1 is every n^th odd number, n>1.

This relationship satisfies the Pythagorean identity, as shown below:

                                                           - Equation 1.

Let us rename the hypotenuse constructed from Table 2 as Minimal Even Leg Hypotenuse or MELH. It is to be noted that the first term in Table 2 does not produce the MELH as the even leg 4 is greater than the odd leg 3. Also, the first hypotenuse produced in Table 2 is the same as the first hypotenuse in Table 1, with two legs interchanged.

Investigation of Probabilities:

Probability of a given number being an MOLH:

As can be inferred from Table 1, for the n^th odd number, we have n-1 MOLHs.

The MOLH corresponding to n is, 2n*(n-1)+1.

If  x is the x^th integer starting from 1 corresponding to the n^th odd number, then

                                                                                                     - Equation 2.

Replacing n with ,

The MOLH corresponding to the x^th integer can be re-written as:

                                                                     - Equation 3.

This means there are n-1 or  MOLHs equal to or less than .

Let .

Then,

                                                                                                                       -Equation 4.

This means, there are  MOLHs less than or equal to m.

Then the probability of a given positive integer m being an MOLH is,

                                                                                       -           -Equation 5.

Probability of a given number being an MELH:

As can be inferred from Table 2, for the n^th even number, the table has n-1 hypotenuses. Out of these, the first hypotenuse cannot be counted as an MELH as the odd leg 3 is smaller than the even leg 4. This means, we have n-2 MELHs for the n^th even number.

The MELH corresponding to n is, n²+1.

If  x is the x^th integer starting from 1 corresponding to the n^th even number, then

                                                                                                              - Equation 6.

Replacing n with ,

The MELH corresponding to the x^th integer can be re-written as:

                                                                                                             -Equation 7.

This means there are n-2 or  MELHs equal to or less than .

Let

Then,

                                                                                                                      -Equation 8.

This means, there are  MELHs less than or equal to m.

Then the probability of a given positive integer m being an MELH is,

                                                                                                  -Equation 9.

From the Tables 1 and 2 as well as Equations 5 and 9, it can be easily inferred that a number being an MELH has a higher probability than being an MOLH.

Fig. 2 below shows the plot of probabilities of a number being MELH vs a number being MOLH. Both the probabilities tend to zero as m tends to infinity.

Fig 2. Probability distribution of a given number being an MOLH/MELH

 Conclusions and Inferences:

·        The Pythagorean triple forming the Minimum Odd Leg Hypotenuse or MOLH is of the form 2n-1, 2n*(n-1), and 2n*(n-1)+1, where 2n-1 is every n^th odd number, n>1.

·        It is known that the Pythagorean triple forming the Minimum Even Leg Hypotenuse or MELH is of the form 2n, n²-1 and n²+1, where 2n is every n^th even number, n>1.

·        All MOLHs are odd. All MELHs are odd and even, alternatively.

·        The probability of a given number m being an MOLH is given by  .

·      The probability of a given number m being an MELH is given by  .

·        MELHs occur more frequently than MOLHs.