I. INTRODUCTION

The theory of numbers is a fascinating area of mathematics. Right integer triangles have attracted the attention of many mathematicians and math enthusiasts because it is a treasure house where finding many hidden connections is like going on a treasure hunt. Refer to [1]-[3] for various fascinating challenges. In addition to polygonal numbers, we also have the Jarasandha numbers, Nasty numbers, Dhuruva numbers, and Canada Numbers, which are all intriguing patterns of numbers. These figures are displayed in [4]-[9]. Special Pythagorean triangles linked to Nasty and polygonal numbers are derived in [10]-[15].

In this writing, we look for a distinguished couple of right integer triangles where the difference in their perimeters in each pair is four times the Canada numbers.

II. BASIC DEFINITIONS

Definition: 1

The ternary quadratic Diophantine equation given by $s^2+t^2=r^2$ is known as Integer right equation, where $s,t$ and $r$ are natural numbers and denotes it by $\mathrm{\Delta }(s,t,r)$. Also, in $\mathrm{\Delta }(s,t,r)$: $s^2+t^2=r^2$, $s$ and $t$ are called its legs and $r$ its hypotenuse.

Definition: 2

The most cited solution of the Integer right equation is

where . If $a$ and $b$ have opposing parities and $gcd(a,b)=1$, then this solution is referred to as primitive.

Definition: 3

Canada numbers are those $n$ such that the sum of the squares of the digits of $n$ is equal to the sum of the non-trivial divisors of $n$, i.e., $\sigma (n)-n-1$.

The Canada numbers are 125, 581, 8549 and 16999.

The name of these numbers is due to the fact they were defined by some mathematicians from Manitoba University to celebrate the ${125}^{th}$ anniversary of Canada.

III. MATERIALS AND METHODS

Let ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ be two distinct right integer triangles with generators , respectively. Let ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ be the perimeters and ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ be the areas of ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ such that ${\mathrm{P}}_1-{\mathrm{P}}_2=4$ times the $1^{\text{st}}$ Canada number 125.

The equation derived from the relationship above is

It is observed after completing numerical calculations that there are 20 different values for $a$, $c$, and $b$ satisfied (1) provided $a+b+c=\text{Canada number}$

The values of $a,c,b,{\mathrm{P}}_1$ and ${\mathrm{P}}_2$ are shown in Table I below for clarity and simplicity.

Table I

Thus, it can be observed that there are 20 couples of right integer triangles, where each couple's difference in perimeters equals four times the first Canada number (125). Of these 20 couples, ten are non-primitive, six are primitive, and four are couples, where one is a primitive triangle and the other is non-primitive.

The following Table II illustrates a similar observation of other Canada numbers:

Table II

IV. CONCLUSION

In this article, we propose a couple of integer right triangles whose perimeter differences are each equal to four times the Canada number. We also provide the number of couples containing primitive and non-primitive integer right triangles. In conclusion, one can look for relationships between distinguished couples of integer right triangles and other unique numbers and number patterns.