Pell-Pythagoras Numbers – Where Geometry Meets Number Theory

                                                                                                                        By Srimahaishnu Vinjamuri

Introduction

The previously discussed article titled “A Tale of Two Legs: Generating Pythagorean Triples from Odd and Even Beginnings” examined the concepts of Minimum Odd Leg Hypotenuses (MOLHs) and Minimum Even Leg Hypotenuses (MELHs). It was observed that only a limited number of hypotenuses are common to both MOLHs and MELHs. This phenomenon was further investigated, yielding noteworthy results.

Fig 1. Pictorial representation of Pythagoras theorem

Definition – Pythagorean Triple:

When a, b and c can be expressed as integers, they are called a Pythagorean triple, with c representing the length the hypotenuse.

In case of triangles where the least leg is an odd number, the three legs a,b,c are of the form  a, (a²-1)/2, (a²⁺1)/2  respectively.

In case of triangles where the least leg is an even number (except for 4,3,5), a,b,c are of the form 2n, n²-1 and n²+1, which is straightforward.

Common Hypotenuses between MOLHs and MELHs:

For an MOLH and MELH to be equal,

 has to be equal to n²+1.

This can be written in equation form as:

This can be further re-written as:

 

Substituting a with x, and n with y

we have

                                          - Equation 1

                                                                            

Equation 1 is the famous Pell’s equation which is of the form

, where m =2.     

The fundamental root of Equation 1 is, x=3 and y=2.

It is known that the subsequent equations can be found recursively using: - Equation 2

This also means

 The established solution from this is:

 - Equation 3

- Equation 4

                                        

The Results:

The initial ten values of x and n are listed in the table below.

Table 1: Values of MOLH and MOEH

k	x	n	Minimum Odd Leg a	Minimum Even Leg 2n	MOLH (a2+1)/2	MOEH n2+1
1	3	2	3	4	5	5
2	17	12	17	24	145	145
3	99	70	99	140	4901	4901
4	577	408	577	816	166465	166465
5	3363	2378	3363	4756	5654885	5654885
6	19601	13860	19601	27720	192099601	192099601
7	114243	80782	114243	161564	6525731525	6525731525
8	665857	470832	665857	941664	221682772225	221682772225
9	3880899	2744210	3880899	5488420	7530688524101	7530688524101
10	22619537	15994428	22619537	31988856	255821727047185	255821727047185

 

The corresponding Pythagorean triples are listed in the table below.

Minimum Odd Leg Triple	Minimum Even Leg Triple
3,4,5	4,3,5
17,144,145	24,143,145
99,4900,4901	140,4899,4901
577,166464,166465	816,166463,166465
3363,5654884,5654885	4756,5654883,5654885
19601,192099600,192099601	27720,192099599,192099601
114243,6525731524,6525731525	161564,6525731523,6525731525
665857,221682772224,221682772225	941664,221682772223,221682772225
3880899,7530688524100,7530688524101	5488420,7530688524099,7530688524101
22619537,255821727047184,255821727047185	31988856,255821727047183,255821727047185

 

It is evident that the first Minimum Even Leg Triple has its odd leg smaller than the even leg.

It can be easily observed and established that the ratio of n and x at each iteration tends  to

.

The hypotenuses grow even more rapidly at the following rate, which is nearly 33.97 times at each iteration for a large hypotenuse.

Let us call these common hypotenuses Pell-Pythagoras numbers. These numbers form the series 5,145,4901,166465,5654885 and so on.

Probabilities:

From my previous article, the following were determined.

The probability of a given positive integer m being an MOLH is,

                      -Equation 5

                     -Equation 6

From the present article, it can be observed that the common hypotenuses between MOLH and MELH are very rare.

Hence, it can be safely concluded for all practical purposes that the probability of a given number being a hypotenuse of a minimum legged Pythagorean triple (MOEH or MOLH) is:

 -Equation 7                                                                                                            

Conclusions:

•  Only a very limited set of Pythagorean triples, those with the shortest legs being both odd and even, share the same hypotenuse. These are rare intersections between Minimum Odd Leg Hypotenuses (MOLH) and Minimum Even Leg Hypotenuses (MELH). This family of shared hypotenuses can be called “Pell–Pythagoras numbers.”, which form the series 5,145,4901,166465,5654885 and so on.
•  It is established that the common hypotenuses of these triples satisfy a specific Pell’s equation of the form x²−2y²=1. This demonstrates a beautiful and unexpected connection between classical geometry (Pythagoras) and algebraic number theory (Pell’s equation).
•  Such intersections are extremely rare. Based on prior probabilistic estimations for MOLH and MELH hypotenuses, the probability that a number is a common hypotenuse is virtually zero in the limit. This rarity is mathematically supported by the exponential growth of solutions to Pell’s equations.