Solving the Pells Equation - Case Study Example

RUNNING HEAD: Pells Equation Pells Equation Name Tutor Date Solving pells equation Pell's equation is the Diophantine equation which has the form the form: - –=1, in this case, n is a non-square integer whilst x and y are both integers. Slightly, x = 1 and y = 0 always can be used to solve this equation. Lagrange was able to prove that for any natural number n, which is not a perfect square, there must be an x and a y>0 which can be able to satisfy pells equation. In addition, infinitely many such kinds of solutions of this type of an equation exist. These solutions give way to good rational estimation of the variety x/y to the square root of n. Pell numbers arise in olden times and most remarkably in rational approximation to the problem of solving the square root of 2. If two integers, that is x and y form an answer to a Pell equation –=1, Then it holds that their ratio presents a secure approximation to. The succession of approximations in this form is then: - , Where the denominator for each and every fraction is a Pell number (the numbers with n=2). The numerator is the addition of a Pell number and its antecedent in the series of Pell numbers. Pell numbers can be defined as: - =0, =1, and then for n>1, =2+, therefore 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378 … That is, the rough calculation to has the form . This type of an approximation was acknowledged by Indian mathematicians in the third or fourth century B.C. the approximations can then be derived from the continued fractions for the expansion of. As much as –=-1 is solvable just for some certain values of D, the technique of continued fractions always provides solutions for if and only if they exist. The condition which is necessary to be solved by –=-1 where D replaces 2 is that the prime factors of D become of the form 4n+1 and it should also hold that D should not be divisible by 4 (doubly even). As demonstrated by Nagell (1951), by the equation , the conditions are not sufficient for existence. In all resulting discussion, pay no concentration to the minor result , . Let indicate the nth convergent, consequently we shall have been capable to work out equations 1 if it is promising to locate an identity that acts in accordance with the equations: - Interestingly, this happens to be a probable way in which this can be solved from the continued fraction and having a surd of the form This surd becomez periodic at a term where . In order to get the ontinued fraction, convergents to use recursive relations as shown below. In this case, x is the floor function. To compute the additional functions and which are defined by: - The most important identities which have been satisfied by the convergent for the continued fractions are therefore: - According to Beiler (1966), both the linear equation and the quadratic are both solved by getting the quadratic fraction that best solves them. If we were to let to be the term at the point which the continued fraction happens to be of a periodic nature, which is normal for quadratic surds, the pell equation , witth r(odd), is said to be positive and the result for the small integers becomes and also for the case where is the convergent of r. if in the case of r being even, is negative though to have the solution put in a form of small integers, and which summarize: This entails that the equation can be analogously be solved in the following manner with +1 on the right hand side if and only if r is even and has no solution for an odd value of r. And also = = This has a variety of solutions: X = Y = The solutions are known to hold for: . This takes into account that n takes odd values only. According o Beiller (1966), the table below will indicate the integer solutions (x, y) to the pell equation with D0 has no solutions. Pells equation connection with the stern brocots tree The stern brocot is generated from 2 fractions 0/1 and 1/0 by repetitively performing the mediant procedure. In its natural form, it associates the sequences (binary) in L and also in R with positive reduced fractions as below: The stern-brocot tree has a close relationship to Euclidian algorithms and continued fractions theory. There is an important form which contains matrices instead of fractions. For the matrices which correspond to a/b fraction has dual columns in immediate left and also the right predecessors of a/b in the unique tree. In this tree, the columns sum up to give the fraction which can be regarded as the vector. For a binary sequence with Ls and Rs can have a direct translation to a certain matrix through the interpretation of L and R to be the erstwhile used matrices and then multiplying them. If N was to be computed in the pells equation, the solution is as same as in the stern brocot tree which includes the addition of the columns to each of the others towards the right or left. N= RL2L . This occurs in the solution to x2-2y2=1 Public key RSA of cryptography In RSA, there are both public and private keys. The public key is open to all for encryption purposes. The private key is used to decrypt the message. The steps involved in this process are three: key generation, encryption and decryption. To generate the key, the following streps are followed: - 1. Choose wo prime integers at random: p and q. 2. Calculate n = pq where n is modulus for both keys. 3. Calculate φ(pq) = (p − 1)(q − 1) where φ is the Euler’s function 4. Choose e for 1 Read More