magupta lemma.

The Diophantine equation x − Dy = 1, also known as Pell’s equation, is known to have an infinite number of integer solutions for any square - free positive integer D > 1 [2]. The solutions stem from a single fundamental solution found in Algebraic Number Theory. Brahmagupta and Bhaskara were the first to study the classical Pell’s equation, and Lagrange developed the entire theory.There are an infinite number of integer solutions (xn, yn) to Pell’s equation x −Dy = ±1 for n ≥ 1. Since all other solutions to this equation can be (easily) derived from the first non-trivial positive integer solution (x1, y1) (in this case, x1 or x1 + y1 D is minimum), it is known as the fundamental solution. Let x − Dy = 1 have a fundamental solution (x1, y1). Then, for any integer n ≥ 2, the n-th positive solution of this equation, (xn, yn), is defined by the equality xn + yn D = (x1 + y1 D) .The corresponding negative Pell’s equation, x − Dy = −1, (1) is considerably more mysterious. For (1) to be solvable in integers, it is required that D ≡ 1 or 2 (mod 4), and that all odd prime divisors of D have forms that are congruent to 1 modulo 4. These conditions, however, do not guarantee the existence of a solution. One may visit A031396 in the OEIS [16] for some known numbers D = 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, ... such that (1) is solvable in integers x and y.For several decades, certain number theorists have been committed to developing standards for the solvability of (1). A few requirements are Qr listed in [2, 10, 11, 17]. According to Newman’s [11] demonstration, (1) can be solved in integers if D = pi, pi where r = 2 or r is odd, and pi’s are primes congruent to 1 modulo 4 and satisfy = −1 for all 1 ≤ i, j ≤ r pj with i = ̸ j. The standard Jacobi symbol is in this case. Conversely, Mollin [10] found a relationship between x − Dy = 1 and the negative Pell’s equation (1).

He demonstrated that (1) can be solved in integers x and y if and only if x0 ≡ −1 (mod 2D) is satisfied by the fundamental solution (x0, y0) of x − Dy = 1. Despite the significance of these findings, their approaches are constrained if D ≫ 1. For instance, there are numerous situations where the fundamental solutions are fairly large. Thus, extensive computations are required to verify these conditions. Positive Pell’s equations have recently been found to have well-known solvability criteria. A primitive Pythagorean triple for D can be found by using the length of its period as a guide. Another method is to compute the simple continued fraction of √ D.[18-20] A prime number that satisfies a result about the prime-counting function proved by Srinivasa Ramanujan is known as a Ramanujan prime in mathematics. As he points out, Bertrand’s postulate was initially proven by Chebyshev. Ramanujan presented a revised demonstration of the concept in 1919. Ramanujan arrived to a generalized result at the end of the two-page published work. Consequently, 2, 11, 17, 29, and 41 are the initial five Ramanujan primes.

Take note that Rn must be a prime number in order for it to exist:π(x) − π and, hence,π(x) must increase x Rn by obtaining another prime at x = Rn. Since π(x) − π can increase by atmost 1,π(Rn) − π = n. In

2 2 t th

this paper the negative Pell’s equation x = (Rp)y − (R1) formed using the p Ramanujan prime Rp, are analysed for obtaining non-zero integer solution for various choice of t and are illustrated for primes less then 100.

Theorem 0.1. If x1, y1 is the fundamental solution of x − Dy = 1,then 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) , n = 1, 2, 3 · · ·

For illustration,consider the fundamental solution x1 = 24 & y1 = 5 of x − 23y = 1. A second positive solution (x2, y2) can be obtained from the formula √ √ √ x2 + y2 23 = (24 + 5 23) = 1151 + 240 23

This gives x2 = 1151, y2 = 240.These integers also satisfy the equation

2 2 2 2

x − 23y = 1,since 1151 − 23y = 1.

∴ Any positive solution can be calculated using the Theorem 2.1.

Theorem 0.2. Suppose that the equation x − Dy = −1 (2) has solution in positive integers and (x0, y0) be its minimal solution.The general solution to (2) is given by (xn, yn)n≥0 where xn = x0un +Dy0vn,yn = y0un +x0vn and (un, vn)n≥0 is the general solution to Pell’s equation u − Dv = 1.

Theorem 0.3. Let p be a prime. The negative Pell’s equation x − py = −1 is solvable if and only if p = 2 or p ≡ 1 (mod 4).

2 2 t th

The equation under consideration in the negative Pell’s equation x = (Rp)y − (R1) , where Rp denotes the p Ramanujan prime and R1 = 2,the first Ramanujan Prime.

System-I: Diophantine equation x = 11y − 2 ,with R2 = 11 and R1 = 2.

This system concerns with negative Pell’s equation x = 11y − 2 , t ∈ N.Here the prime R2 = 11 cannot be expressed as the sum of two relatively prime squares.Thus by the condition (2.1.1) of the algorithm,the negative Pell’s equation x − 11y = −1 is not soluble. Thus we can conclude that the negative Pell’s equation x = 11y − 2 is insoluble.

System-II: Diophantine Equation x = 17y − 2 with R3 = 17, R1 = 2.

The prime R3 = 17 satisfies all the conditions of Theorem 2.3,and hence we conclude that the negative Pell’s equation x = 17y − 2 ,t ∈ N is solvable and infinitely many positive integer solutions are obtained for the various choices of t.The more general Pell’s equation u = 17v + 1 (I) can be viewed and its general solution (un, vn) is given by

un = fn; vn = √ gn, (II)

Choice-I: t is an even number Case 1:t=2, The Pell’s equation under consideration is

with the initial solution x0 = 8; y0 = 2.Applying Theorem 2.2, a series of non-zero integer solutions to (3) are discovered as

yn = √ [2 17fn + 8gn] ,where fn and gn are defined as in (II)

The recurrence relations satisfied by the solutions of (3) are given by

yn+2 − 66yn+1 + yn = 0 Case 2:t = 2k + 2, where 1 ≤ k ≤ 7, The Pell’s equation thus obtained is with initial solution x0 = 2 , y0 = 2 .Applying Brahmagupta lemma between the solutions(x0, y0) and (un, vn) from (II),the sequence of non-zero distinct integer solutions to (4) are obtained as

xn = [fn + 17gn]

yn = √ [ 17fn + gn].

Case 3: t=2k+16, k ∈ N

with initial solution x0 = 59(2 ), y0 = 125(2 ) Applying Theorem 2.2, and using (II) we get infinite number of solutions to (5) as

Choice-II: t is an odd number Case 1: t = 2k + 1, 1 ≤ k ≤ 4 The Pell’s equation to be solved is

The following table represents various values of t with the initial & general solutions to (6).

t = 2k + 1 (x0, y0) (xn, yn) 1 √ 1 √ k = 1 (3,1) xn = [3fn + 17gn] ; yn = √ [ 17fn + 3gn] 1 √ 1 √ k = 2 (6,2) xn = [6fn + 2 17gn] ; yn = √ [2 17fn + 6gn] 1 √ 1 √ k = 3 (5,3) xn = [5fn + 3 17gn] ; yn = √ [3 17fn + 5gn] 1 √ 1 √ k = 4 (10,6) xn = [10fn + 6 17gn] ; yn = √ [6 17fn + 10gn] Case 2: t=2k+9, k ∈ N The Pell’s equation thus obtained is

Let (x0, y0) be the initial solution of (7)given by x0 = 3(2 ), y0 = 11(2 ).

Applying Theorem 2.2, a series of non-zero integer solution to (7) are discovered as

yn = √ [11 17fn + 3gn], where fn and gn are defined as in (II).

The recurrence relations satisfied by the solutions of (7) are given by

yn+2 − 66yn+1 + yn = 0 System -III: Diophantine Equation x = 29y − 2 with R4 = 29, R1 = 2.

Given that R4 = 29 satisfies all the settings of Theorem 2.3,we can conclude that the negative Pell’s equation x = 29y − 2 is solvable in integers.

Consider the case when t is an even number.

In this case,the Pell’s equation is

2 2 2k

Let (x0, y0) be the primary solution of (8) specified by x0 = 5(2 ), y0 = 2 . Analysing the pell’s equation u = 29v + 1 to discover the additional solutions to (8), whose generic solution (un, vn) is provided by un = fn; vn = √ gn

Applying Brahmagupta lemma between the solutions(x0, y0) and (un, vn),the sequence of non-zero distinct integer solutions to (8) are obtained as

The recurrence relations fulfilled by the solutions of (8) are specified by

yn+2 − 1960yn+1 + yn = 0 Observation:This system fails to generate integer solutions for all the odd choices of t.

System-IV:Diophantine Equation x = 41y − 2 with R5 = 41, R1 = 2.

For this particular equation,we consider the prime 41.Using the Algorithm and testing with (a, b) = (4, 5),−bv + 2avw + bw = 1 has a solution (v, w) = (2, 1).Therefore,we can develop this discussion by confirming that the negative pell equation x = 41y − 2 .t ∈ N is solvable in integers.

Consider the pellian equation u = 41v + 1 (III) whose general solution (un, vn) is given by

un = fn; vn = √ gn (IV)

Choice-1: t is an even number

The Pell’s equation to be checked for its solutions is

Assume (x0, y0) be the fundamental solution of (9) given by x0 = 64, y0 = 10 Applying Theorem 2.2, a collection of nonzero different integer solutions to (9) is achieved as The recurrence relation derived by employing the solutions are

yn+2 − 4098yn+1 + yn = 0.

Case2: t = 2k + 2, 1 ≤ k ≤ 3

Let (x0, y0) represent the first solution to (10) proposed by x0 = 5(2 ), y0 = 2 Applying Brahmagupta Lemma between (x0, y0) and (un, vn) from (IV),the series of different non-zero integer solutions are discovered as

Case 3: t = 2k + 8, k ∈ N The Pell’s equation is

Let (x0, y0) represent the initial solutions to (11) offered by x0 = 2 , y0 = 5(2 ) Applying Brahmagupta Lemma between (x0, y0) and (un, vn) from (IV),the sequence of non-zero distinct integer solutions to (11) are obtained as

xn = [fn + 5 41gn]

yn = √ [5 41fn + gn].

Choice-2: t is an odd number

Applying Theorem 2.2 ,a collection of nonzero different integer solutions to (12) are achieved as

Case 2: t = 2k + 3, k ∈ N The Pell’s equation to be examined is

Applying Brahmagupta Lemma between (x0, y0) and (un, vn) from (IV),the sequence of non-zero distinct integer solutions to (13) are obtained as

In this paper, the equations are all formed restricting the primes less than 100. It can be extended to any bound and also to any prime number and thus delve into the world of Pell’s equations.

Credit authorship contribution statement:

Manju Somanath: Visualization, Validation,Conceptualisation.

V.Sangeetha: Writing – original draft, Validation, Data curation.

T.Anupreethi: Writing – original draft, Visualization,Validation.

Declaration of competing interest:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.