One more method to generate Pythagorean Triples K.L. Ramakumar

INTRODUCTION

Any mathematics student knows Pythagorean Theorem and Pythagorean triples. Pythagorean triples are any three numbers a, b, c such that square of the largest number is equal to sum of the squares of the other two numbers. For example the three numbers 3, 4, and 5 are Pythagorean triples as 32 + 42 = 52. Given below are simple procedures to find out Pythagorean triples either from an even or from an odd number. To find out Pythagorean triples from a given even number, I will define a theorem: Theorem: Square of any even number is always divisible by 4. In other words, 4 is always one of the factors of square of any even number. Proof: Any even number N is either divisible by 4 or can be expressed as (M ± 2) where M is another even number divisible by 4. (e.g., 22 = 20 +2 or 22 = 24 – 2. Hence M = 20 or 24)) Let N be an even number not divisible by 4. Now N2 = N x N = (M ± 2) x (M ± 2) = M2 ± 2x2xM + 22 As M is divisible by 4, M x M is also divisible by 4. The other two terms are clearly divisible by 4 as they contain 4 as coefficient. Therefore the algebraic sum of all the terms in the expression is divisible by 4. Hence square of an even number N is always divisible by 4. PYTHAGOREAN TRIPLES FROM A GIVEN EVEN NUMBER (MORE THAN 2) Given below is a simple method, which I found very useful: Step-1: Select any even number greater than or equal to 4. Step-2: Take the square of the number. Step-3: Divide the product thus obtained by 4. Step-4: Subtract 1 to get one of the Pythagorean triples. Step-5: Add 1 to get the other triple number. e.g.1: Let the number be 4. To get the remaining two Pythagorean triples: Square of 4 = 4 x 4 = 16 16/4 = 4 Second Pythagorean triple = 4-1 = 3 Third Pythagorean triple = 4 + 1 = 5 So the three Pythagorean triples are 3, 4, 5

e.g.2: Let the number be 8. The remaining two Pythagorean triples are: Square of 8 = 8 x 8 = 64 64/4 = 16 Second Pythagorean triple = 16 – 1 = 15 Third Pythagorean triple = 16 + 1 = 17 The Pythagorean triples are 8, 15, 17 Check: 8 x 8 + 15 x 15 = 64 + 225 = 289 = 17 x 17 Selecting 2 gives trivial result. Let us see. 2 x 2 = 4 4/4 = 1 1-1 = 0: 1 + 1 = 2 So the numbers are 0, 2, 2 PYTHAGOREAN TRIPLES FROM AN ODD NUMBER For completion sake, I am giving a method below, to find out Pythagorean triples from an odd number. Select any odd number. Case 1 Take the square of the number. Divide the product thus obtained into two numbers in such a way that the difference between them is 1. Subtract 1 from the product. Divide by 2. Quotient is one number and quotient + 1 is another number You get a set of Pythagorean triples e.g. Let the number be 11 Square of 11 is 121 Divide 121 into two numbers. How? (121-1 = 120: 120/2 = 60. Add 1 to 60 to get 61) Therefore the Pythagorean triples are 11, 60, 61

Check 11 x 11 + 60 x 60 = 121 + 3600 = 3721 = 61 x 61 Case 2 If the selected odd number is not a prime (A prime number is solely divided only by itself other than 1), then we can have more than one set of Pythagorean triples for that number.

Find out the factors for this number. You will get small odd numbers. Follow the procedure given in Case 1. You will end up different sets of Pythagorean triples. e.g. Let the number be 15. Using the procedure given in case 1,we get a set of Pythagorean triples as 15, 112, 113 Now find out the factors for 15. These are 3 and 5 For 3, using the procedure given in case 1, the Pythagorean triples are 3, 4, 5. Multiply each number by 5 to get another set of Pythagorean triples as 15, 20, 25. For 5, using the procedure given in case 1, the Pythagorean triples are 5, 12, 13. Multiply each number by 3 to get yet another set of Pythagorean triples as 15, 36, 39. We, therefore get three sets of Pythagorean Triples with one of the numbers as 15. These three sets are: (a) 15, 112, 113 (b) 15, 20, 25 (c) 15, 36, 39 Check the result for correctness. I leave it to the readers. After finding out the Pythagorean triples as mentioned above, multiply each number by any other number to get another set of Pythagorean triples. e.g. Let the first set of Pythagorean triples are 3, 4, 5 Multiplying by 2 we get 6, 8, 10 Multiplying by 3 we get 9, 12, 15 Multiplying by 4 we get 12, 16, 20 etc. All the above are Pythagorean triples.

Any set of Pythagorean triples, when multiplied by any same number will give another set of Pythagorean triples.

Thus we have two types of Pythagorean triples. One category, we can call them primary Pythagorean triples and are generated using (i) the procedure outlined for even numbers or (ii) the procedure mentioned in case 1 for odd numbers. The other category, we can call them secondary Pythagorean triples and are generated from primary Pythagorean triples by multiplication with a number. To summarize, we can state the following:

1. If any given number ‘a’ is even, then the Pythagorean triples are

a, a

2

14

, a

2

14

OR a a

a, ,

2 2

1 12 2

2. If any given number ‘a’ is odd, then the Pythagorean triples are

a, a a

, 2 21 12 2

GENERAL PROPERTIES OF PYTHAGOREAN TRIPLES Any set of Pythagorean triples, when multiplied by any same number will give

another set of Pythagorean triples. There will be only one set of Pythagorean triples if the smallest among them is a

prime number. If the smallest of the Pythagorean triples is a prime number then the other

numbers in the set are successive numbers. If the smallest of the Pythagorean triples is not a prime number but an odd

number, then one set of Pythagorean triples has two successive numbers. There are no Pythagorean triples with smallest member as 2 or 4.

All primary Pythagorean triples with even number as the smallest, has the other

two members of the set differing by 2. Any Pythagorean triple set contains at least one even number.