Many mathematicians have investigated the properties of continued fractions. They made fraction expansions Pi number, golden ratio and many more special numbers. With help fractions, solutions some Diophantine equations are obtained. In this study, encryption was using fractional square root non-perfect-square integers. Each 29 letters in alphabet is represented by nonperfect integers starting ...

Abstract By using a jump transformation associated to the Romik map, we define new continued fraction algorithm called odd-odd fraction, whose principal convergents are rational numbers of odd denominators and numerators. Among others, it is proved that all best approximating rationals numerators an irrational number given by vice versa.

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σM of reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σM for M → ∞ were provided by Hensley [12]. In the rational case, first studies by Cusick, Hensley and Vallée [4, 9, 19] considered the case of a ...

We detail the continued fraction expansion of the square root of monic sextic polynomials. We note in passing that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In particular we obtain an associated Somos sequence defined by a three-term recurrence relation of width 6.