Perfect powers that are sums of consecutive squares

We determine all perfect powers that can be written as the sum of at most 10 consecutive squares

Perfect powers that are sums of squares in a three term arithmetic progression

We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.Comment: 6 page

On perfect powers that are sums of two Fibonacci numbers

We study the equation $F_n + F_m = y^p$, where $F_n$ and $F_m$ are respectively the $n$-th and $m$-th Fibonacci numbers and $p \ge 2$. We find all solutions under the assumption $n \equiv m \pmod{2}$.Comment: 6 page

A Lucas–Lehmer approach to generalised Lebesgue–Ramanujan–Nagell equations

From Springer Nature via Jisc Publications RouterHistory: received 2020-01-22, accepted 2021-02-03, registration 2021-02-03, online 2021-06-10, pub-electronic 2021-06-10, pub-print 2021-11Publication status: PublishedAbstract: We describe a computationally efficient approach to resolving equations of the form C1x2+C2=yn in coprime integers, for fixed values of C1, C2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier

On the difference between permutation polynomials over finite fields

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p>(d^2-3d+4)^2$, then there is no complete mapping polynomial $f$ in \Fp[x] of degree $d\ge 2$. For arbitrary finite fields \Fq, a similar non-existence result is obtained recently by I\c s\i k, Topuzo\u glu and Winterhof in terms of the Carlitz rank of $f$. Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if $f$ and $f+g$ are both permutation polynomials of degree $d\ge 2$ over \Fp, with $p>(d^2-3d+4)^2$, then the degree $k$ of $g$ satisfies $k \geq 3d/5$, unless $g$ is constant. In this article, assuming $f$ and $f+g$ are permutation polynomials in \Fq[x], we give lower bounds for $k$ in terms of the Carlitz rank of $f$ and $q$. Our results generalize the above mentioned result of I\c s\i k et al. We also show for a special class of polynomials $f$ of Carlitz rank $n \geq 1$ that if $f+x^k$ is a permutation of \Fq, with $\gcd(k+1, q-1)=1$, then $k\geq (q-n)/(n+3)$

Two year review of maternal mortality at a tertiary care hospital of GMERS, Valsad, Gujarat, India

Background: According to the WHO, 80 of maternal deaths in developing countries are due to direct maternal causes such as haemorrhage, hypertensive disorders and sepsis. These deaths are largely preventable. Maternal mortality ratio (MMR) in India is 167/100,000 live births.Methods: This retrospective observational study was conducted at GMERS, Valsad. Data regarding maternal deaths from January 2016 to December 2017 were collected and analyzed with respect to epidemiological parameters. The number of live births in the same period was obtained from the labour ward ragister. Maternal mortality rate and Mean maternal mortality ratio for the study period was calculated.Results: The mean Maternal mortality rate in the study period was 413.3/100,000 births. The maternal mortality ratio (MMR) in India is 167/100,000 live births. More than half of maternal deaths were reported in multiparous patients. More maternal deaths were observed in women from rural areas (67.3%), unbooked patients (73.3%) and illiterate women (65.3%). Thirty six (69.3%) maternal death occurred during postpartum period. Most common delay was first delay (60.0%) followed by second delay (40.0%). Postpartum haemorrhage (28.8%), preeclampsia (17.3%), sepsis (13.46%) were the major direct causes of maternal deaths. Indirect causes accounted for one third of maternal deaths in our study. Anemia, hepatitis and heart disease were responsible for 13.4%, 5.7%, and 1.9% of maternal deaths, respectively.Conclusions: Majority of maternal deaths are observed in patients from rural areas, unbooked, and illiterate patients. Hemorrhage, eclampsia and sepsis are leading causes of maternal deaths. Most of these maternal deaths are preventable if patients are given appropriate treatment at periphery and timely referred to higher centers