the classical formulas produce all primitive triples they do not generate all possible triples, especially ... be generated using a set of linear tran...

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Abstract: The method of generating Pythagorean triples is known for about 2000 years. Though the classical formulas produce all primitive triples they do not generate all possible triples, especially non-primitive triples. This paper presents a direct method to generate all possible triples both primitive and non-primitive for any given number. Then the technique is generalized to produce Pythagorean Quadruples and n-tuples. Our method utilizes the fact that the difference between lengths of the hypotenuse and one leg of a Pythagorean triangle can have only certain distinct values depending on the length of the other which remains true for higher tuples also.

I.

Introduction

A Pythagorean triple is an ordered triple of positive integers

such that

(1) An effective way to generate Pythagorean triples is based on Euclid’s formula found in his book Elements. This formula states that for any two positive integers and with form a Pythagorean triple. Though this classical formula generates all primitive triples and infinitely many of them, it is incapable of generating all the triples including non-primitive cases. For example the triple cannot be generated from the formula rather introducing a multiplier to the triple does so. Besides we observe that while Euclid’s formula produces the triple , it doesn’t produce ; a transposition is needed. Also by a result of Berggren (1934) all primitive Pythagorean triples can be generated using a set of linear transformations but it requires the triple to start with. So we theorise a direct method to generate all possible primitive and non-primitive triples for a given number (one leg of a right angle triangle). Our strategy will be the observation that the difference between and (or between and ) can have only certain distinct values depending on the given number (or ). Let us consider . So that equation (1) becomes

(2) (3)

1

Equation (3) clearly shows that must be a factor of for integral values of . This is the first constraint that prevents possessing any arbitrary value. Further we note that b to have positive value

(4) This is the second condition imparted on . In sections II and III we describe the case of Pythagorean triples extensively, in section IV we extend the theory for quadruples and finally in section V we discuss the generalization to n-tuples. Section VI provides a simple technique to generate tuples of arbitrary length starting from any single number.

II.

Generating Primitive Pythagorean Triple

A Pythagorean triple is said to be primitive when are coprime to each other i.e. gcd . In this section we discuss for a given value of what are the possible values of . Before we proceed let us recall some basic properties of primitive triple. We know is always odd and only one of and is odd, so if we choose as even will be even and if we set as odd will also be odd. Now numbers can be divided into three categories on the basis of their prime factorization. (A) even numbers which are only powers of . (B) odd numbers consisting of powers of any prime. (C) even numbers containing both powers of and other primes. So we divide our discussion into three subsections: (A) Let

and

Equation (4) demands that

, where

and

are whole numbers.

. Now from equation (2) we get

(5) Since in this case

is even

must be odd. Now the right hand side of equation (5) will be odd only

if . Hence the triple will be

.

Example: Let . So . Hence, and . Thus we get which is a primitive triple. But if we choose , say we obtain and . Clearly the triple non-primitive that can be obtained from primitive triple . So, in this case the only possible value of is .

2

is

(B) Let , where is a prime factor of can have any positive integral value. Then (2), we get

and is the product of other prime factors and where is also a whole number. From equation

(6) Now for the triple to be primitive must not have that either or provided . Hence the triple will be Here

for

as a factor. So, from this equation it is clear or

for

.

ensures us that for each odd number there exists at least one primitive triple of the

form

.

Example: Let us consider . Then can have two values, For , , . Clearly forms a primitive triple. Next for , , which again makes a primitive triple In this case we also observe that if we take , we should have possible because it violates condition (4). (C) Let , where is one prime factor of and and can have any positive integral values. Then From equation (2) we get

or

.

. but it is not

is the product of other prime factors , where are whole numbers.

(7) (i) When , the terms within the parentheses on right hand side of equation (7) give odd number so that will be odd only if and also must not be a factor of for the triple being primitive that requires or . So, the primitive triple will be for and for . Example: Let us consider . For , ,

. Then

can have two values,

and

. So the primitive triple is .

For primitive. (ii) When

,

,

. This gives

which is also

, equation(7) becomes

(8) 3

Again the terms within the parentheses on right hand side of equation (8) give odd number so that will be odd only if and also should not have as a factor for the triple being primitive which requires or . Example: Let us consider and . For , , primitive triple. For , , primitive.

. Then

will have two values

. So we get

which is a

. This gives

which is also

In this case if we choose instead of it would give not permissible since becomes greater than . (iii) When

which is

, from equation(7) (9)

, will always be even which gives non-primitive solutions. The only primitive triples are obtained when but this will lead to equation (6) which we discuss If

earlier.

Here one important fact to be noted is that when solution so that integers of the configuration triple. Example: The numbers

, we never obtain a primitive will always give non-primitive

will always form non-primitive triple.

etc. If we represent the given number as ∏ with or ,

then

will be of the form

where

.

A general example: Let us consider show the different cases in the following table.

. Here

and

Table 1: Primitive triples for

-----

-----

-----

-----

4

. We

-----

-----

-----

-----

Here the last four values of are not possible because of the condition (4). So only four primitive triples can be generated for . Thus following the previous rules all possible primitive triples for a given number can be generated.

III.

Generating Non-primitive Pythagorean Triple

In the process of generating non-primitive Pythagorean triple the only constraint is . So we first need to factorize the given number and then will be any combination of those factors except the cases for primitive triples. So it is obvious that if is even must be even and if is odd will also be odd. We illustrate the method by the following example. Let

. Again we show various cases in the following table. Table 2: Non-primitive triples for

Here and have not been taken because they generate primitive triples and the other combinations of the factors have been discarded as they violate condition (4). Thus finding all the possible values of

we can obtain all non-primitive triples.

5

IV.

Pythagorean Quadruple

A Pythagorean quadruple is an ordered quadruple of positive integers

such that (10)

In this section we discuss how to generate all possible Pythagorean quadruples for a given set of . Let and . Then equation (10) becomes

(11) Three facts are clear from equation (11): (i) (ii) (iii)

if is even must be even and if is odd must be odd for integral value of . when is even it ought to be an integral multiple of . for positive value of it is required that (12)

We are mainly interested in the generation of primitive quadruples which we will discuss in three sections. Case (A): Here

is even and

is odd (or

is an odd number, so

(i) We first consider that represented as

is even and

is odd)

is odd. and

have common factors , then and will be of the form and

can be

all being integers for all . Equation (11) then

gives (13) So for primitive solution from

to

or

; and

with the restriction given by equation (12). Then

or Example: (1) So but Thus we get,

for all and and

can take all integral values ∏

∏

, with either

has those values discussed above.

. Then is not possible as

So the primitive quadruples for (2) Let So

,

.

and and and

when when are

and . Other combinations of 6

and

. Then are not possible due to equation (12).

.

Table 3: Primitive Pythagorean quadruples for

(ii)

Now we consider that ∏

where

Example: Let

and

have no common factors so that

can take all integral values from and

and

to

. Then

provided .

Here we draw an important conclusion that whenever one of one primitive quadruple with . and

. .

Table 4: Primitive Pythagorean quadruples for

Case (B): Both

, then

and

or

is odd, there will be at least

are even

Here is an even number, so will be even. This case will be same as discussed in case (A) except the introduction of some power of so that now and . Equation (11) then gives (14) So, the conditions for obtaining primitive quadruples are same with an additional condition either or . Then the values of are ∏ value from

∏ to

∏

or

, provided

∏

, here

or

and

has any integral

.

Again an important conclusion from this theory is that for any two given even numbers there always exists a primitive quadruple with . Example: Let and following table shows all possible primitive cases.

7

. Then

. The

Table 5: Primitive Pythagorean quadruples for

and

, as their squares are greater than . Case (C): Both

and

are odd

To study this case we represent integers. Then

and

as

and

where ,

are positive

(15) Since is even must be even but then must contain a factor for being an integral multiple of . But equation (15) clearly shows that is a multiple of only i.e. . So no set of quadruples (primitive and non-primitive) can be obtained if both the given numbers are odd. To produce non-primitive quadruples, we first calculate and factorise it. Then combination of those factors expect those for primitive cases with the condition Following examples will make it clear. Example: (i) Let Then

and . So the values of

can have any .

. that give non-primitive quadruples will be

. Table 6: Non-primitive Pythagorean quadruples for

and

Here give primitive quadruples and the rest combinations violate condition (12), so they are not taken.

8

(ii) Let primitive cases are

and

. The

. Here the

Table 7: Non-primitive Pythagorean quadruples for

and

Here give primitive sets and the rest do not obey the condition (12). So these are not taken.

In this way all Pythagorean Quadruples can be generated for any two given numbers.

V.

Pythagorean n-tuple A Pythagorean n-tuple is a set of n positive integers

such that (16)

When ( with

numbers are given, we can calculate the rest two and form an n-tuple . This process is quite similar to the process or generating primitive quadruples and .

Now among given

numbers some will be odd, say

numbers and the rest will be even.

(i) When , . This does not lead to the formation of n-tuple which is similar to case (C) of quadruple. (ii) When , is odd too and with some powers are common factors of the given ( ,

or

and

, where ) integers. Then

can be any integer from

to

satisfying

. Example: Let .

,

,

,

and

, .

9

,

,

and

Table 8: Primitive Pythagorean n-tuples (n=10) for 8 given numbers

So the primitive n-tuples (n=10) for those 8 numbers are

(iii) When (greater than 2) or i.e. all the given numbers are even, then is also even and it will have the form , , where or some powers of them are common factors of the given numbers and or

and

are other primes. Then

∏

∏

can take any integral value from

to

provided

Example: Let Now primitive solutions.

,

,

,

,

,

,

or

,

and

.

.

. So

for

Table 9: Primitive Pythagorean n-tuples (n=9) for 7 given numbers

So the primitive n-tuples (

) for the given

10

numbers.

We can generate the non-primitive n-tuples also in a manner similar to the case of quadruple discussed previously.

VI.

Generating Pythagorean n-tuple starting from a single number

Having the discussion of generating different Pythagorean tuples elaborately here we give a simple method to generate tuples of arbitrary length starting from a given number using the theory of Pythagorean triple. If the number is given, we can calculate the triple ( ) i.e. starting from , ( ) can be obtained. So the equation extends to quadruple. Proceeding in the same way we can elongate the chain and after obtain the n-tuple: .

. Then ,a iterations we

Now it is to be noted that for a given number ( ) we can have several triples, so each of them will form a different branch and each branch will have a sub-branch and so on. A branch will be primitive only if gcd of any three numbers is 1. In this way we can obtain several n-tuples from a given number but this method is incapable of generating all possible cases since the method totally depends on the process of generating triples. Example: Here we show three branches with

.

VII. Conclusion The major advantage of our method is that it does not require any primitive set to start with and finding proper multipliers or transformations to obtain the desired tuple. An interesting fact is that just by factorizing we can forecast how many primitive and non-primitive cases are possible before actually calculating them which cannot be done by classical formulas. We are also able to produce tuples of any length starting from a given number in a very simple way.

References Books: [1] Jagadguru Swami Sri Bharati Krisna Thirthaji Maharaja (1981). Vedic Mathematics. Indological Publishers and Booksellers [2] Gareth A. Jones and J. Mary Jones (2005). Elementary Number Theory. Springer Internet Resources: http://en.wikipedia.org/wiki/Pythagorean_triple http://en.wikipedia.org/wiki/Pythagorean_quadruple http://en.wikipedia.org/wiki/Pythagorean_triple#Pythagorean_n-tuple 11