Positive Nonsquare Integer Research Articles

On the Integer Solution of the Transcendental Equation

Let C be a positive non-square integer. In this paper, we look at the complete solutions of the Transcendental equation , where 𝒙𝟐−𝑪𝒚𝟐=𝜶𝟐 or 𝟐𝟐𝒕. In addition, we find repeated relationships in the solutions to this figure.

ON THE FACTORISATION OF

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p\nmid D$ and $0<\unicode[STIX]{x1D70E}<0.847$. We show that there exist effectively computable constants $C_{1}$ and $C_{2}$ such that if there is a solution to $x^{2}+D=p^{n}$ with $p^{n}>C_{1}$, then for every $x>C_{2}$ with $x^{2}+D=p^{n}m$ we have $m>x^{\unicode[STIX]{x1D70E}}$. As an application, we show that for $x\neq \{5,1015\}$, if the equation $x^{2}+76=101^{n}m$ holds, then $m>x^{0.14}$.

Expressions for rational approximations to square roots of integers using Pell's equation

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation: p2 − Dq2 = 1, where D is any positive non-square integer. With known positive integer solution p1 and q1 we can compute, using these recursive expressions, pn and qn for all n > 1. See Table in [2] for a list of smallest integer, or fundamental, solutions p1 and q1 for D≤ 128. These (pn, qn) pairs also form rational approximations to that, as noted in [3, Chapter 3], match with convergents (Cn = pn / qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for .

Continued fractions expansion of √D and Pell equation x2 - Dy2 = 1

Let D ≠ = 1 be a positive non-square integer. In the first section, we give some preliminaries from Pell equations and simple continued fraction expansion. In the second section, we give a formula for the continued fraction expansion of √D for some specific values of D and then we consider the integer solutions of Pell equations x2-Dy2 = 1 for these values of D including recurrence relations on the integer solutions of it.

The Pell Equation X<sup>2</sup> - Dy<sup>2</sup> = ± k<sup>2</sup>

Let D≠1 be a positive non-square integer and k≥2 be any fixed integer. Extending the work of A. Tek-can, here we obtain some formulas for the integer solutions of the Pell equation X2 - Dy2 = ± k2 .

Powers in recurrence sequences: Pell equations

In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the form x 2 - D y 2n = 1 in integers x, y and n > 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.

Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation

In this paper, we refine work of Beukers, applying results from the theory of Pade approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to \(\sqrt m\) (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have $$\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}$$ provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.

A note on a theorem of Ljunggren and the Diophantine equations x 2 - kxy 2 + y 4 = 1, 4

Let D denote a positive nonsquare integer. Ljunggren has shown that there are at most two solutions in positive integers (x, y) to the Diophantine equation x 2–Dy 4 = 1, and that if two such solutions (x 1, y 1), (x 2, y 2) exist, with x 1≤x 2, then $x_1+y_1^2\sqrt {D}$ is the fundamental unit $\epsilon _{D}$ in the quadratic field ${\Bbb Q}(\sqrt {D})$ , and $x_2+y_2^2\sqrt {D}$ is either $\epsilon _{D}^2$ or $\epsilon _{D}^4$ . The purpose of this note is twofold. Using a recent result of Cohn, we generalize Ljunggren’s theorem. We then use this generalization to completely solve the Diophantine equations x 2–kxy 2 + y 4 = 1, 4.

Some results on pseudosquares

If p p is an odd prime, the pseudosquare L p L_p is defined to be the least positive nonsquare integer such that L p ≡ 1 ( mod 8 ) L_p\equiv 1\pmod {8} and the Legendre symbol ( L p / q ) = 1 (L_p/q)=1 for all odd primes q ≤ p q\le p . In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L 271 L_{271} . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that L p > e p / 2 L_p> e^{\sqrt {p/2}} , an inequality that must hold under the extended Riemann Hypothesis.

Comparison of the lengths of the continued fractions of √𝐷 and \frac12(1+√𝐷)

Let D D denote a positive nonsquare integer such that D ≡ 1 ( mod 4 ) D \equiv 1\,(\bmod 4) . Let l ( D ) l(\sqrt D ) (resp. l ( 1 2 ( 1 + D ) ) l(\tfrac {1} {2}(1 + \sqrt D )) ) denote the length of the period of the continued fraction expansion of D \sqrt D (resp. 1 2 ( 1 + D ) ) \tfrac {1} {2}(1 + \sqrt D )) ). Recently Ishii, Kaplan, and Williams (On Eisenstein’s problem, Acta Arith. 54 (1990), 323-345) established inequalities between l ( D ) l(\sqrt D ) and l ( 1 2 ( 1 + D ) ) l(\tfrac {1} {2}(1 + \sqrt D )) . In this note it is shown that these inequalities are best possible in a strong sense.

Pell's equations X2 − mY2 = −1, −4 and continued fractions

Let m denote a positive nonsquare integer. It is shown that if Pell's equation X 2 − mY 2 = −1 is solvable in integers X and Y then the equation X 2 − mY 2 = −4 is solvable in coprime integers X and Y if and only if l(√m) ≡ l( 1 2 (1+√m)) (mod 4), where l( α) denotes the length of the period of the continued fraction expansion of the quadratic irrational α.

An Analogue of the Nearest Integer Continued Fraction for Certain Cubic Irrationalities

Let $\theta$ be any irrational and define $Ne(\theta )$ to be that integer such that $|\theta - Ne(\theta )|\; < \frac {1}{2}$. Put ${\rho _0} = \theta$, ${r_0} = Ne({\rho _0})$, ${\rho _{k + 1}} = 1/({r_k} - {\rho _k})$, ${r_{k + 1}} = Ne({\rho _{k + 1}})$. Then the r’s here are the partial quotients of the nearest integer continued fraction (NICF) expansion of $\theta$. When D is a positive nonsquare integer, and $\theta = \sqrt D$, this expansion is periodic. It can be used to find the regulator of $\mathcal {Q}(\sqrt D )$ in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field $\mathcal {F}$ with negative discriminant $\Delta$. This new algorithm (NIVCF) is periodic and can be used to find the regulator of $\mathcal {F}$. If $I < \sqrt [4]{{|\Delta |/148}}$, the NIVCF algorithm can be used to find any algebraic integer $\alpha$ of $\mathcal {F}$ such that $N(\alpha ) = I$. Numerical results suggest that the NIVCF algorithm finds the regulator of $\mathcal {F} = \mathcal {Q}(\sqrt [3]{D})$ in about 80 percent of the time needed by Voronoi’s algorithm.

An analogue of the nearest integer continued fraction for certain cubic irrationalities

Let θ \theta be any irrational and define N e ( θ ) Ne(\theta ) to be that integer such that | θ − N e ( θ ) | &gt; 1 2 |\theta - Ne(\theta )|\; &gt; \frac {1}{2} . Put ρ 0 = θ {\rho _0} = \theta , r 0 = N e ( ρ 0 ) {r_0} = Ne({\rho _0}) , ρ k + 1 = 1 / ( r k − ρ k ) {\rho _{k + 1}} = 1/({r_k} - {\rho _k}) , r k + 1 = N e ( ρ k + 1 ) {r_{k + 1}} = Ne({\rho _{k + 1}}) . Then the r’s here are the partial quotients of the nearest integer continued fraction (NICF) expansion of θ \theta . When D is a positive nonsquare integer, and θ = D \theta = \sqrt D , this expansion is periodic. It can be used to find the regulator of Q ( D ) \mathcal {Q}(\sqrt D ) in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field F \mathcal {F} with negative discriminant Δ \Delta . This new algorithm (NIVCF) is periodic and can be used to find the regulator of F \mathcal {F} . If I &gt; | Δ | / 148 4 I &gt; \sqrt [4]{{|\Delta |/148}} , the NIVCF algorithm can be used to find any algebraic integer α \alpha of F \mathcal {F} such that N ( α ) = I N(\alpha ) = I . Numerical results suggest that the NIVCF algorithm finds the regulator of F = Q ( D 3 ) \mathcal {F} = \mathcal {Q}(\sqrt [3]{D}) in about 80 percent of the time needed by Voronoi’s algorithm.

An upper bound for the period of the simple continued fraction for √D

Let p(D) denote the length of the period of the simple continued-fraction expansion of VD, where D is a positive non-square integer. In this paper, it is shown that p(D) Ί, and an estimate for p(D) is given when D is not squarefree. 1* Introduction* The problem of finding a good upper bound for the length p(D) of the period of the simple continued fraction for Ί/D, where D is a positive nonsquare integer, has received relatively little attention. Recently, Hickerson [6] and Hirst [7] have given estimates for p(D); Hickerson's estimate implies that

Problems on Periodic Simple Continued Fractions

Let N be a positive non-square integer and a(1),a(2),...,a(3) be the partial denominators in the period of length s = s(N) of the continued fraction for radicalN. Also let Sigma(N) = a(3) - a(3-1) + -... +/- a(1), and let h(d) be the class-number of Q( radicald). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime identical with3(4) and p > 3, then h(p) = 1 implies that Sigma(p) = 3h(-p).This result led to related conjectures presented herein.