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).
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
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.
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 .
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.