Journal of Mathematics
Volume 2019 (2019), Article ID 4286517, 8 pages
https://doi.org/10.1155/2019/4286517

Research Article

Pythagorean Triples with Common Sides

Raymond Calvin Ochieng,¹ Chiteng’a John Chikunji,² and Vitalis Onyango-Otieno¹

¹Strathmore Institute of Mathematical Sciences, Nairobi, Kenya
²Department of Biometry and Mathematics, Botswana University of Agriculture and Natural Resources, Gaborone, Botswana

Correspondence should be addressed to Raymond Calvin Ochieng; rochieng@strathmore.edu

Received 16 January 2019; Accepted 6 March 2019; Published 1 April 2019

Academic Editor: Ji Gao

Copyright © 2019 Raymond Calvin Ochieng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

1. Introduction

A Pythagorean triple (PT) is a triple of positive integers , which satisfy the Pythagorean equation

where represents the length of the hypotenuse; and represent the lengths of the other two sides (legs) of a right triangle. We say a Pythagorean triple is primitive if the numbers , , and are pairwise coprime (see [1]).

Several methods have been formulated that generate Pythagorean triples (see [2–9]). For instance, the most common one is the classical Euclid formula [10, 11]:

whenever . A triple generated by this method is primitive if and only if and is odd. Note that is odd if , have opposite parity [12], or

Now by Euclid’s general formula, any Pythagorean triple, primitive and nonprimitive triples, can be written as , where is some positive integer and , are as defined in [10, 11].

Pythagorean triples form different patterns that can be classified and applied in various fields such as cryptography; see [13–20]. For instance, there exist Pythagorean triples that have identical legs; e.g., and are two primitive triples with , while and are nonprimitive triples with the same leg. Similarly , , , and are four primitive Pythagorean triples which have as the identical leg. The nonprimitive triples which share the leg are discussed in Example 1.

In [1], Sierpinski states and proves that there exist only a finite number of Pythagorean triples with a given leg He further states and proves that, for each positive integer , there exist at least different Pythagorean triples with the same leg , where For instance, if we take

where , then we obtain Pythagorean triples with the same leg and with different hypotenuses.

It is also stated in [1] that it is not easy to prove that, for each positive integer , there exist at least different primitive Pythagorean triples with an identical leg. However in this paper we prove formula for determining pairs of primitive Pythagorean triples which have identical legs. We also show how to determine all the primitive and nonprimitive Pythagorean triples which have a given identical leg.

2. Pairs of Pythagorean Triples with Identical Leg

Consider the following pairs of primitive Pythagorean triples which can be generated from the equation

From Table 1, we observe that, for each pair of triples with identical leg, the difference between the hypotenuse and the odd leg is either or We then have

Solving (5), we obtain

Observe from Table 1 that is the form where Substitute this to obtain

In a similar way, we can obtain and from (6).

Table 1: Pairs of primitive Pythagorean triples with identical leg, even.

3
2
12
5
13

6
1
12
35
37

5
2
20
21
29

10
1
20
99
101

7
2
28
45
53

14
1
28
195
197

9
2
36
77
85

18
1
36
323
325

We have thus derived the following.

Proposition 1. If is an even leg of the Pythagorean triple , then the following pair of equations produce primitive Pythagorean triples with identical leg :

for all

Proof. We show the first equation in (9) is a Pythagorean triple.

We then show it is a primitive Pythagorean triple. By Euclid’s formula,

is a primitive Pythagorean triple if , , and , are integers of opposite parity.
Now,

To solve for and , add (13) and (14), to obtain

for all
Subtract (13) from (14) to get

Clearly ; for all , and

Therefore (9) is a primitive triple.
The last equation in (9), that is, , can be shown in a similar manner.

Now, consider pairs of PPTs with identical leg, odd, and, similarly, these can be obtained from (2).

From Table 2, we have two cases considering the difference between the hypotenuse and the leg

Table 2: Pairs of primitive Pythagorean triples with identical leg, odd.

4
1
15
8
17

8
7
15
112
113

5
2
21
20
29

11
10
21
220
221

6
1
35
12
37

18
17
35
612
613

8
3
55
48
73

28
27
55
1512
1513

Case I. If , we have

Observe, from Table 2, that is a semiprime; that is, where and are primes,

Case II. In this case, for some prime , we have

Substitute in both cases to obtain the following.

Proposition 2. Let be the odd leg of a Pythagorean triple , and then

produce a pair of primitive Pythagorean triples with identical odd leg , for all odd primes , with

Proof. Now

that is, the first equation in (20) is a Pythagorean triple. We then show it is primitive. Since both and are odd primes, let

Add (22) and (24) to get

and subtract (22) from (24) to obtain

But and are odd primes so and for some , with Substitute and in and to obtain

It follows that the first equation in (20) is a primitive Pythagorean triple for and are consecutive positive integers and, hence, are of opposite parity and
The second equation in (20) can be shown in a similar way.

In [21, 22], a formula is given which determines the number of primitive Pythagorean triples that have a common leg. However, these formulas do not show how to obtain the primitive Pythagorean triples. In Proposition 6, we show how to determine all the primitive as well as nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

For easy reference, we first state and prove the following.

Lemma 3 (see [21, 22]). Consider the triple of positive integers with as the even leg. Then the number of primitive Pythagorean triples with as a common leg is

where is the number of prime divisors of

Proof. If is a primitive Pythagorean triple and is even, then we have integers and such that , , , and Each such pair uniquely determines Since is even, there is no solution unless Suppose , and suppose is even, without loss of generality. If denotes the set of prime divisors of , any subset of uniquely determines , and hence , since no prime can divide both and There are choices of , and hence as many choices of expressing in the form with and

Lemma 4 (see [21, 22]). Let be a Pythagorean triple with as the odd leg. Then the number of primitive Pythagorean triples with as a common leg is given by

where is the number of prime divisors of Also

Proof. We wish to count the number of positive integer pairs , such that with , , and The parity of forces both factors , to be odd, so that , are of opposite parity. Moreover Choosing the prime factors for one of , determines the prime factors of the other, and , are uniquely determined from , However since we must reserve the larger factor of for , only half of all the subsets count.

The following lemma will be useful in the proof of the proposition below.

Lemma 5. Consider the Pythagorean triple with as the even leg. is not primitive Pythagorean triple if any of the following hold:

(1)if is odd(2)if where is odd(3)if

Proof. If is a primitive Pythagorean triple, then we have integers and such that , , , and If is odd, we contradict If where is odd, then and are both odd, but then If , then , a contradiction.

We extend the Lemmas 3 and 4 to determine all the primitive and nonprimitive Pythagorean triples that have a common leg, either odd or even.

Proposition 6. Let be a Pythagorean triple. Define where is the number of prime divisors of , and where for some such that , is the number of prime divisors of for which is a primitive Pythagorean triple. Then the number of primitive and nonprimitive Pythagorean triples that have a common leg is

Proof. Suppose the leg is odd, then is a primitive Pythagorean triple if , and Then is of the form

where are odd primes and
By factorization,

We solve for and such that where , , and for all pairs in which and Each of these pairs corresponds to a primitive Pythagorean triple. By Lemma 4, if is the number of prime divisors of , then the number of primitive Pythagorean triples with common leg is given by
If is a nonprimitive triple then , , and have a greatest common divisor The possibilities of are the factors of We eliminate for the case when , that is, the case when This is because is the leg of a Pythagorean triangle and should satisfy .
Let the remaining cases of be for some For each , is a primitive Pythagorean triple when

for , , and
Let the number of prime divisors of be , for each Then there are primitive Pythagorean triples with a common leg, Each of these triples is then multiplied by to obtain nonprimitive Pythagorean triples with the leg
The number of all the nonprimitive Pythagorean triples for all is given by

Then all primitive and nonprimitive Pythagorean triples are given by where and are as defined.
Now, suppose that the leg is even, then is a primitive Pythagorean triple if , , and The leg is of the form

where are odd primes and
Now the set of generating pairs of positive integers, that have opposite parity, are relatively prime and are

That is, by Lemma 3, if denotes the set of prime divisors of , any subset of uniquely determines , and hence , since no prime can divide both and There are choices of , and hence as many choices of expressing in the form with .
Suppose is a nonprimitive Pythagorean triple. Then , , and have a greatest common divisor The possibilities of are drawn from the factors of We consider all such that is primitive. Two cases arise.
Case I. By Lemma 5, we eliminate any such that is odd; where is odd and Moreover, as the even leg of a Pythagorean triangle,
Let the remaining values of be for some For each , is a primitive Pythagorean triple when

for , , and
We then find all pairs such that Each of these pairs of produces a primitive Pythagorean triple, which is then multiplied by to produce a nonprimitive triple with leg as desired.
Case II. We consider such that , odd. Then and we proceed to determine the primitive triples as described above, for odd case of The primitive triples obtained are then multiplied by to obtain nonprimitive triples with a leg equal to
The number of these nonprimitive Pythagorean triples for each is where is the number of prime divisors of The sum of all these triples is

Then the number of all primitive and nonprimitive Pythagorean triples is

as desired, where and are as defined above.

We illustrate this with some examples.

Example 1. The proof of Proposition 6 lays out an easy way of determining all the Pythagorean triples, both primitive and nonprimitive, that have an identical leg.
Consider the Pythagorean triples with an identical leg If is a primitive Pythagorean triple, then Solve for , if , where and such that We obtain , which produce the four primitive Pythagorean triples shown in Table 3.
These numbers of primitive triples produced agree with Lemma 4; that is,

Now suppose is nonprimitive, then , , and have a greatest common divisor The possibilities of are In this case, we only eliminate , that is, the case when We then consider the remaining values of
Let Then is primitive if

where such that and Now But , so that These produce the nonprimitive triples:

In a similar way, leads to the triples If , we have Finally, each of , , and , respectively, leads to the triples , and
Observe the number of nonprimitive Pythagorean triples described for each in Table 4.

Table 3: Primitive Pythagorean triples with common leg

53
52
105
5512
5513

19
16
105
608
617

13
8
105
208
233

11
4
105
88
137

Table 4: Number of nonprimitive Pythagorean triples with common leg for each value of .

3
35
2
2

5
21
2
2

7
15
2
2

21
5
1
1

35
3
1
1

Total

8

Example 2. Let This implies and by considering its factors, the possible set of pairs of are

These produce the eight primitive Pythagorean triples shown in Table 5.
Suppose is nonprimitive, then , , and have the greatest common divisor The possibilities of are

Two cases arise.
In the first case, by Lemma 5, we eliminate all such that is odd, where is odd and Moreover These conditions exclude the following set .
We consider each of the remaining cases of , that is,
Let , and then is primitive when

where such that and Now and the possibilities satisfying the conditions above are These produce the primitive triples

Multiply each of these primitive triples by to obtain

as desired.
In a similar way the other values of lead to the nonprimitive triples shown in Table 6.
Secondly, we consider such that is odd. Then .
Let , and then and as such , and This has been solved in Example 1. We obtain , which produce triples as in Table 7.
Note that when , nonprimitive triples are generated as well. However we find that such triples will be produced in the remaining values of For example, primitive triples generated when are and , which on multiplying by leads to and , respectively. But and are, respectively, multiples of and , which are nonprimitive triples when As such to avoid repetition, we only take the primitive triples that arise from each .

Table 5: Primitive Pythagorean triples with common leg .

210
1
420
4099
44101

105
2
420
11021
11029

70
3
420
4891
4908

42
5
420
1739
1789

35
6
420
1189
1261

30
7
420
851
949

21
10
420
341
541

15
14
420
29
421

Table 6: Nonprimitive Pythagorean triples with common leg for each odd value of .

Primitive Triple
Non-primitive Triple

5
(42, 1)
(84, 1763, 1765)
(420, 8815, 8825)

(21, 2)
(84, 437, 445)
(420, 2185, 2225)

(14, 3)
(84, 187, 205)
(420, 935, 1025)

(7, 6)
(84, 13, 85)
(420, 65, 425)

7
(30, 1)
(60, 899, 901)
(420, 6293, 6307)

(15, 2)
(60, 221, 229)
(420, 1547, 1603)

(10, 3)
(60, 91, 109)
(420, 637, 763)

(6, 5)
(60, 11, 61)
(420, 77, 427)

15
(14, 1)
(28, 195, 197)
(420, 2925, 2955)

(7, 2)
(28, 45, 53)
(420, 675, 795)

21
(10, 1)
(20, 99, 101)
(420, 2079, 2121)

(5, 2)
(20, 21, 29)
(420, 441, 609)

35
(6, 1)
(12, 35, 37)
(420, 1225, 1295)

(3, 2)
(12, 5, 13)
(420, 175, 455)

105
(2, 1)
(4, 3, 5)
(420, 315, 525)

Table 7: Nonprimitive Pythagorean triples with common leg for each even value of .

Primitive Triple
Non-primitive Triple

4
(53, 52)
(105, 5512, 5513)
(420, 22048, 22052)

(19, 16)
(105, 608, 617)
(420, 2432, 2468)

(13, 8)
(105, 208, 233)
(420, 832, 932)

(11, 4)
(105, 88, 137)
(420, 352, 548)

12
(18, 17)
(35, 612, 613)
(420, 7344, 7356)

(6, 1)
(35, 12, 37)
(420, 144, 444)

20
(11, 10)
(21, 220, 221)
(420, 4400, 4420)

(5, 2)
(21, 20, 29)
(420, 400, 580)

28
(8, 7)
(15, 112, 113)
(420, 3136, 3164)

(4, 1)
(15, 8, 17)
(420, 224, 476)

60
(4, 3)
(7, 24, 25)
(420, 1440, 1500)

84
(3, 2)
(5, 12, 13)
(420, 1008, 1092)

140
(2, 1)
(3, 4, 5)
(420, 560, 700)

Repeat the process for other values of to obtain the triples shown in Table 7.

We see, from Tables 5–7, that there are primitive and nonprimitive Pythagorean triples with as a common leg.

3. Conclusion

Propositions 1 and 2 define infinitely many pairs of primitive Pythagorean triples that have identical legs. It remains an open problem for one to extend the formulas in these propositions and generalize for any number of primitive Pythagorean triples which have identical legs.

Proposition 6 presents a simple technique of finding all the primitive and nonprimitive Pythagorean triples associated with a given value, the leg of a Pythagorean triple. This makes it easy to classify Pythagorean triples, with respect to the values of the legs, and investigate different properties that might be of interest to a researcher. For instance, one can easily classify Pythagorean triples by considering the divisibility of legs by any positive integer or legs of Pythagorean triples that satisfy some sequences, among many other properties.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was done while the first author was conducting his Ph.D. research at Strathmore University, and he would like to acknowledge the University for financial support.

References

W. Sierpinski, Pythagorean Triangles, Dover Publications, Inc., Mineola, NY, New York, USA, 2003.

J. Rukavicka, “Dickson's method for generating Pythagorean triples revisited,” European Journal of Pure and Applied Mathematics, vol. 6, no. 3, pp. 363-364, 2013.

A. F. Horadam, “Fibonacci number triples,” The American Mathematical Monthly, vol. 68, pp. 751–753, 1961.

F. R. Bernhart and H. L Price, Heron’s Formula, Descartes circles, and Pythagorean triangles, 2005, https://arxiv.org/abs/math/0701624.

H. L. Price, The Pythagorean Tree: A New Species, 2008, https://arxiv.org/abs/0809.4324.

J. L. Poet and J. Vestal, “Curious consequences of a miscopied quadratic,” The College Mathematics Journal, vol. 36, no. 4, pp. 273–277, 2005.

A. Carvalho and C. P. Santos, “A very useful Pythagorean tree,” in Proceedings of the Recreational Mathematics Colloquium II, University of Vora, Portugal, April 27–30, 2011, J. N. Silva, Ed., pp. 3–15, Associao Ludus, Lisboa, 2012.

P. W. Wade and W. R. Wade, “Recursions that produce Pythagorean triples,” The College Mathematics Journal, vol. 31, no. 2, pp. 98–101, 2000.

D. McCullough and E. Wade, “Recursive enumeration of Pythagorean triples,” The College Mathematics Journal, vol. 34, no. 2, pp. 107–111, 2003.

D. M. Burton, Elementary Number Theory, McGraw-Hill, New York, NY, USA, 2002.

W. J. Spezeski, “Rethinking pythagorean triples,” Applications and Applied Mathematics. An International Journal (AAM), vol. 3, no. 1, pp. 100–112, 2008.

D. W. Mitchell, “An alternative characterisation of all primitive pythagorean triples,” The Mathematical Gazette, vol. 85, no. 503, pp. 27–35, 2001.

M. Gnanaguruparan, “Recursive hiding of secrets in visual cryptography,” Cryptologia, vol. 26, no. 1, pp. 68–76, 2002.

S. Kak, Pythagorean Triples and Cryptographic Coding, 2010, https://arxiv.org/abs/1004.3770.

S. Kak and M. Prabhu, “Cryptographic applications of primitive pythagorean triples,” Cryptologia, vol. 38, no. 3, pp. 215–222, 2014.

Y. Kothapalli, Indexing Properties of Primitive Pythagorean Triples for Cryptography Applications, 2011, https://arxiv.org/abs/1101.4236.

A. Luma and B. Rauf, “Data encryption and decryption using new pythagorean triple algorithm,” in Proceedings of the World Congress on Engineering 2014, vol. I, WCE 2014, London, UK, 2014.

K. Mani and A. Devi, “Modified DES using different keystreams based on primitive pythagorean triples,” International Journal of Mathematical Sciences and Computing, vol. 3, no. 1, pp. 38–48, 2017.

A. Overmars and S. Venkatraman, “Pythagorean-platonic lattice method for finding all co-prime right angle triangles,” International Journal of Computer and Information Engineering, vol. 11, no. 11, 2017.

T. Lakshmanan and M. Muthusamy, “A novel secure hash algorithm for public key digital signature schemes,” International Arab Journal of Information Technolog, vol. 9, no. 3, 2012.

T. A. Jenkyns and D. McCarthy, “Integers in Pythagorean triples,” Bulletin of the Institute of Combinatorics and Its Applications, vol. 4, pp. 53–57, 1992.

A. Tripathi, “On Pythagorean triples containing a fixed integer,” The Fibonacci Quarterly, vol. 46/47, no. 4, pp. 331–340, 2008/09.