On the prime divisors of elements of a D(−1)D(-1) quadruple

We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it is not prime and that x is divisible by at least two distinct odd primes. Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1} cannot be extended to a D(-1) quadruple

New upper bounds for Ramanujan primes

For $n\ge 1$, the $n^{\rm th}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\ge R_n$, the interval $(\frac{x}{2}, x]$ has at least $n$ primes. We show that for every $\epsilon>0$, there is a positive integer $N$ such that if $\alpha=2n\left(1+\dfrac{\log 2+\epsilon}{\log n+j(n)}\right)$, then $R_n< p_{[\alpha]}$ for all $n>N$, where $p_i$ is the $i^{\rm th}$ prime and $j(n)>0$ is any function that satisfies $j(n)\to \infty$ and $nj'(n)\to 0$

A binary quadratic approach to X2+(2k−1)Y=kZX^2+(2k-1)^Y=k^Z

A conjecture of N. Terai states that for any integer $k>1$, the equation $x^2+(2k-1)^y =k^z$ has only one solution, namely, $(x, y, z) = (k-1, 1, 2).$ Using the theory of representations of integers by binary quadratic forms, we present methods to verify this conjecture in the case when $k\equiv 0 \pmod 4$, with $2k-1$ a prime power, and in the case when $k$ is any odd integer. We present several values of $k$ for which our method shows that the conjecture is true, while existing methods do not apply. However for $k\equiv 0 \pmod 4$ our method does not always work

ANTICONVULSANT PROPERTIES OF SOME MEDICINAL PLANTS- A REVIEW

Introduction: Epilepsy is the tendency to have seizures that start in the brain. The brain uses electrical signals to pass messages between brain cells and when these signals are disrupted, it leads to a seizure. A number of synthetic antiepileptic drugs are available in practice, but various medicinal plants act as an important source of treatment for epilepsy; plants such as Aeollanthus suaveolens, Passiflora caerulea, Persea americana, Annona diversifolia, and Boerhavia diffusa have good anticonvulsant activity.Objective: Anticonvulsant drugs are used to control the convulsions by inhibiting the discharge and then producing hypnosis. The objective is to understand various medicinal plants and plant components, which are being used as an anticonvulsant.Results: A. suaveolens essential oils are the main constituents were deemed to display anticonvulsant activity. P. caerulea is reputed to have herbal activity as a sedative and anticonvulsant and it is often used as a relatively disease resistant root stock. Whereas P. americana, extract produces its anticonvulsant effect by enhancing gamma-aminobutyric acid ergic neurotransmission and or action in the brain. B. diffusa consists of a calcium channel antagonist compound, liriodendrin that is responsible for its anticonvulsant activity.Conclusion: Since epilepsy has become a common brain disorder, having knowledge of the medicinal plants with an anticonvulsant activity will be beneficial to the society.Keywords: Antiepileptic, Aeollanthus suaveolens, Passiflora caerulea, Persea americana, Annona diversifolia, Boerhavia diffusa

A note on Dujella\u27s unicity conjecture

Using properties of binary quadratic Diophantine equations, we prove that if (r=p^{m} q^{n}), where (p, q) are distinct odd primes and (m, n) are positive integers, then the equation (x^{2}-left(r^{2}+1right) y^{2}=r^{2}) has at most one positive integer solution ((x, y)) with (y lt r-1)

D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES

A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a r. A particular instance yields the result that if r=p(p+2) is a product of twin primes, where p ≡ 1 (mod 4), then the D(-1)-pair {1, 1+r2} cannot be extended to a D(-1)-quadruple. Dujella\u27s conjecture states that there is at most one solution (x, y) in positive integers with y < k-1 to the diophantine equation x2-(1+k2)y2=k2. We show that the Dujella conjecture is true when k is a product of two odd primes. As a consequence it follows that if t is a product of two odd primes, then there is no D(-1)-quadruple {1, b, c, d} with d=1+t2

New hyperchaotic system with single nonlinearity, its electronic circuit and encryption design based on current conveyor

Nowadays, hyperchaotic system (HCSs) have been started to be used in engineering applications because they have complex dynamics, randomness, and high sensitivity. For this purpose, HCSs with different features have been introduced in the literature. In this work, a new HCS with a single discontinuous nonlinearity is introduced and analyzed. The proposed system has one saddle focus equilibrium. When the dynamic properties and bifurcation graphics of the system are analyzed, it is determined that the proposed system exhibits the complex phenomenon of multistability. Moreover, analog electronic circuit design of the proposed system is performed with positive second-generation current conveyor. In addition, an encryption circuit is designed to demonstrate that the proposed system can be used in various engineering applications

Melanoma of Unknown Primary Presenting as a Single Back Mass

In this report, we present a case involving the discovery of metastatic melanoma within a mid-right back mass with the clinical presentation of an epidermoid cyst, but the histological qualities of a lymph node. A 43-year-old male presented with a 5 cm x 5 cm cyst-like mass on his mid-right back that had become painful over the last year and consequently underwent three surgical procedures. First, initial excision of the back mass and histological examination resulted in a diagnosis of metastatic melanoma without epidermal involvement. This was followed by re-excision of the back mass site and sentinel node excision, and finally, lymph node dissection of the right axilla. Of the lymph nodes examined, the sentinel node in the right axilla alone showed evidence of melanoma. The absence of a primary lesion or any histological evidence of regression in a presumed primary site resulted in a diagnosis of melanoma of unknown primary, or occult primary melanoma. To our knowledge, this is the first documented case of an occult primary melanoma presenting as a single mass representing a lymph node in the back