Pythagorean triples

Inherent Properties of Pythagorean Triples
Abstract:

The Pythagorean triples are interesting to everyone because of its well known geometrically related ( right angle triangle) mathematical properties ( sum of two squares is equal to a square) There are many hidden mathematical properties associated with Pythagorean triples. It is a recreational attempt to find out such hidden properties. 

Key words: Pythagorean triple – numeral relation – multi-coincidence

Introduction

Pythagoras, the Greek philosopher and mathematician  formulated his famous theorem around 520 BC.He was the first to predict the geometrical aspect of the mathematical relation of a square equal to a sum of two squares. This oldest theorem still has many mathematical funs. Some of the fascinating and curious properties  associated with Pythagorean triples are predicted here.

Property-1

 If  (a_1,b₁,c₁) and (a₂, b₂, c₂) are any two set of irreducible Pythagorean triples  then c₁ c_{2 ±} ( a ₁ b_{2  ±}  a₂ b₁ ) is always a perfect square provided the two product terms have the same parity (even).

(3,4,5) and (5,12,13) gives,

 5x13 + (3x12 – 4x5) = 81 = 9²

5x13 -  (3x12 – 4x5) = 49 = 7²

Where the two squares 7² and 9²  are in arithmetic progression with  the product c₁ c₂

5 x 13 + (3x12 + 4x5 ) = 121 = 11²

5x13 – [3x12 + 4x5 } =  9 = 3²

 Where the two squares 3²  and 11² are in arithmetic progression with c₁ c₂

This can be explained with the knowledge of Taylor’s numbers for a prime set of Pythagorean triple. Accordingly the two irreducible sets can be expressed  as [(m₁² – n₁² ), 2m₁n₁ , (m₁² + n₁²)] and [(m₂² – n₂² ), 2m₂n₂ , (m₂² + n₂²)]  respectively Substituting these  algebraic values in c₁ c_{2  ±} ( a ₁ b_{2 ±-}  a₂ b₁ ) we get,

(m₁² + n₁²) (m₂² + n₂²)  ± [2m₁n₁ (m₂² – n₂² ) ± 2m₂n₂ (m₁² + n₁²) }

     = [m₁ (m₂  -/+ n₂ )  ± n₁ ( m_{2 +/-} n₂ )]²

It is a perfect square for all possible values of m_1,m_2, n₁ and n₂

Property-2

If  (a_1,b₁,c₁) and (a₂, b₂, c₂) are any two set of Pythagorean triples  then its linear combination gives a relation where sum of two squares is equal to sum of two other squares.

(a₁a₂  + b₁b₂)² + (a₁b₂ – b₁ a₂)²  = (a₁a₂  - b₁b₂)² + (a₁b₂ +b₁ a₂)²  = (c₁c₂)²

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For example, (3,4,5) and (5.12,13) give

63²  + 16²  =  33²  + 56²  =  65²

When  the same Pythagorean triple is used for linear combination, we get higher order Pythagorean triples. e.g., (3,4,5) gives (7,24,25)which in turn gives (336,527,625), where the greatest number of the resultant triple is always a square of the  greatest number in the initial triple.

The product of squares of the greatest numbers in any two Pythagorean triples can be shown to be a sum of four squares.

(c₁c₂)²   = ( a₁² + b₁² ) (a₂² + b₂² ) = (a₁a₂)²  + (a₁ b₂)²  + (b₁a₂)²  + (b₁b₂)²

For example (3,4,5) and (5,12,13) give  65²  = 15²  + 36²  = 20² + 48²

Instead of splitting both the squares c₁ and c_{2 ,} one can split any one of the squares  to get multi-coincidence in Pythagorean relation

(c₁c₂)²   = ( a₁² + b₁² ) c₂² =   c₁² (a₂² + b₂²) 

With the same set of Pythagorean triples, we have, 65²  = 39²  + 52²   = 25²  + 60²                  

Property-3

One can arrive at the numeral relation with sum of two squares is equal to sum of two other squares instantly from any known Pythagorean triple. If (a,b,c) is a known Pythagorean triple, then (am + cn)² + (bm + cn)²  = (cm +an + bn)² + (an-bn)², where m and n may have any integer value. For example with (3,4,5)

m= 1 , n= 1 ;  9²  + 8²  = 12²  + 1²

m= 2 ,n= 3;  21² + 23²   = 31²  + 3²

m=3 , n = 2; 19² + 22²  = 29²   + 2²

Property-4

If (a,b,c) is a Pythagorean triple, then (a+bc)² + (b-ac)²  =  (a-bc)²  + (b+ac)²  = c²  + (c²)² = c² ( c² + 1) . It gives equal sum of two squares in three different ways..For example

(3,4,5)   gives 23²  + 11² = 17²  + 19² = 5²  + 25²= 25 x 26

 (5,12,13) gives  161²  + 53²  = 151²  + 77²  = 13²  + 169²  = 169 x 170

(8,15,17) gives 263² + 121²  = 247² + 151²  = 17²  + 289²  = 289x290

Property-5

Intramural mixing in a given Pythagorean triple is a rich source of generating numeral relations for equal sum of two squares in two different ways. This technique is valid only for numeral relations where equal number of squares with equal number of digits are equated. If the number of squares in one side is less by

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one or more from that in  the other side, then one or more  0² ( for single digit numbers)  00² (for two digit numbers ) must be added without affecting the balance of the relation. The root numbers having unequal number of digits can be changed to have equal number of digits by annexing 0 in front of the numbers. For example in the case of (3,4,5) we have

                                   3²  + 4²  = 5²   + 0²

In intramural mixing the root numbers in side are suffixed or prefixed with the root number in the other side. On can interchange the pairing of numbers ,but once paired , it must be same for both side.

Suffixing with a pair :           35²  + 40²  = 53²  + 4²

Interchanging the pairing  30²  + 45²  = 54²  + 3²

This in turn can be used to generate more and more numeral relations

35²  + 40²  = 53²  + 04²  gives, 5335²  + 440²  = 3553² + 4004²

                                                                         435²  + 5340² =  4053²  + 3504²

 30²  + 45²  = 54²  + 3² gives  5430² + 345² = 3054²  + 4503²

                                                330² + 5445² = 4554² + 3003²

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