Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down. Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}]$.Comment: 13 page

The irrationality of a number theoretical series

Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$

Prenatal Sonographic Diagnosis of Jeune Syndrome

Asphyxiating thoracic dysplasia (Jeune syndrome) is characterized by a narrow thoracic cage, which causes severe respiratory failure with frequent perinatal death; brachymelia, predominantly of the rhizomelic type; renal anomalies; and characteristic radiographic findings of ribs, pelvis, and long tubular bones. Inheritance is autosomal recessive. Prenatal sonographic examination was performed at 17 and 19 weeks of a fetus of parents whose first child had died of Jeune syndrome. The length of the humeri, femora, and tibiae was short (below the mean) for gestational age, and the thorax was abnormally flat and narrow. The iliac wings were square-shaped. We concluded that the fetus had Jeune syndrome. The characteristic skeletal changes of Jeune syndrome are distinct enough at 17 weeks of fetal age to permit sonographic diagnosis