“Number Theory” has time and again thrown its magic at the human world, specifically the world of wizards; the Mathematicians. We all know about the beautiful instances of the prime numbers. In this article, we explore the realm of truncated primes. We are talking about left truncated primes. The initial thought was given to these patterns by “Leslie Card” who first started off by researching into right truncated primes. This was picked up later and subsequent research was done to understand the left truncated primes too. It threw light on the specific part of prime numbers and was given to understand that we have 4260 left truncated primes. The following sequence of left truncatable primes will enable us to understand the magic of these numbers. These numbers are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ... As noted in the sequence, we have another manifestation of the Pell’s equation which is the prime focus of this research. We took up the left truncatable primes and attempted to find solutions for the equation represented by this prime. In this instance, we identified the Pell’s equation x = 137y − 37 , m ∈ ℕ ; using the left truncated prime numbers 137 and 37. Note that 137 is the initial prime number, where upon truncating the left most digit; we get 37. The equation is built on these two primes 137 and 37. Our attempt will be to search for its non-trivial integer solutions. To derive the solutions, we will set the proof rolling by we approached the quest with the case of choices of m generalized in all even and odd integers [1–3].

Theorem 1. If x1 , y1 is considered as the fundamental solution of x − dy = 1. Then to be noted is that every positive solution of the equation is given by xn, yn where xn and yn are the integers determined from xn + yn√d = (x1 + y1√d) , for n = 1,2,3, … [4].

Theorem 3. The negative Pell’s equation x − 137y = −37 , m ∈ ℕ is solvable in integers.

Proof. Consider the negative Pell’s equation x = 137y − 37 , m ∈ ℕ. For the negative Pell’s equation, we will consider the prime p = 137, which satisfies the identified conditions of Theorem 2. Therefore, equation x − 137y = −1, is solvable and we can substantiate with certainty the proof that the negative Pell’s equation x = 137y − 37 , m ∈ ℕ is solvable and prevalent in integers [4], [8].

Using the Algorithm as illustrated in solubility of the negative Pell’s equation for (a, b) = (4,11): −bV +

2 2 2

2aVW + bW = 1 has a solution (V, W) = (−7, 10), hence the negative Pell’s equation x − 137y = −1 is soluble.

Choice 1. m = 1 In our quest to find the other solutions of Eq. (1), consider the generalized form of the Pell’s equation

x = 137y + 1. (2)

The initial solution of Eq. (2) is (6083073,519712) and the general solution (x ̃ n, y ̃ n) given by Theorem 1 as

1 1 n+1 n+1

x ̌ n = fn, y ̃ n = gn, where fn = (6083073 + 519712√137) + (6083073 − 519712√137) ,

2 2√137

gn = (6083073 + 519712√137) − (6083073 − 519712√137) , n = 0,1,2 … .

By applying Brahma Gupta lemma between (x0, y0) and (x ̃ n, y ̃ n) the possible sequence of non-zero distinct integer solutions to Eq. (1) are obtained as given below

xn+1 = x0x ̃ n + dy0y ̃ n , yn+1 = x0y ̃ n + y0x ̃ n, (3) xn+1 = [10fn − √137gn], yn+1 = [−√137fn + 10gn]. (4)

2 2√137

Also to be noted is the recurrence relation satisfied by the solution of Eq. (1) given by

Choice 2. m = 3.

Let (x0, y0) be the initial solution of Eq. (6) given by x0 = 370, y0 = −37. Applying Brahma Gupta lemma between (x0, y0) and (x ̃ n, y ̃ n) the possible sequence of non-zero distinct integer solutions to Eq. (6) are obtained by equation Eq. (3) as given below

Choice 3. m = 5.

integer solutions to Eq. (9) obtained by equation Eq. (3) as Choice 4. m = 2k, k ∈ ℕ.

2 2 2k (12)

x = 137y − 37 , k ∈ ℕ.

Let (x0, y0) be the initial solution of equation Eq. (12) given by x0 = 1744(37) ; y0 = 149(37) .

integer solutions to Eq. (12) are obtained by Eq. (3) as given below

Choice 5. m = 2k + 5, k ∈ ℕ.

2 2 2k+5 (15)

x = 137y − 37 , k ∈ ℕ.

k−1 k−1

Let (x0, y0) be the initial solution of the Eq. (15) given by x0 = 506530(37) , y0 = 50653(37) .

Applying Brahma Gupta lemma between (x0, y0) and (x ̃ n, y ̃ n) the sequence of non-zero distinct integer solutions to Eq. (15) are obtained by Eq. (3) as

xn+1 = ቂ506530fn − 50653ඥ137gnቃ,

yn+1 = ቂ−50653ඥ137fn + 506530gnቃ.

2ඥ137

We conclude this research of solving a Pell’s equation involving the left truncatable primes 137 and 37. As seen above illustrated by the analysis and the steps thereof; we have successfully proved the equation x = 137y − 37 , m ∈ ℕ for all integers. Finding the solution for the Pell’s equation constituted by having left truncated primes reinforced the possibility of having solutions to our identified equation [9], [10].

The authors declare that no external funding was received for the research, authorship, or publication of this article.

This study is based on theoretical analysis and does not involve any experimental or empirical data. All information supporting the findings is included within the article. Further clarifications can be obtained from the corresponding author upon reasonable request.

The authors affirm that there are no known conflicts of interest associated with this publication.