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Anitha Srinivasan

Publications and source records attributed to Anitha Srinivasan.

7 recordsLinked to original sources
arXiv2026

Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

We study the Diophantine equation $\displaystyle{\tfrac{a+1}{b} + \tfrac{b+1}{a} \ = \ k}$, where $k$ is an integer. Using Vieta jumping, we completely classify all positive integer pairs $(a, \, b)$. We prove that the associated integer value $k$ can only be $3$ or $4$. The corresponding solution pairs $(a,\,b)$ are related to the classical Fibonacci numbers. As a consequence, the quantity $\frac{a+b}{\gcd(a, \,b)^2}$ takes only the values $1, \, 2, \, 3$ and $5$. This reveals an unexpected connection between a simple rational Diophantine condition, Vieta jumping, and Fibonacci numbers.

math.NT↗
arXiv2026

Branches of Markoff $m$-triples with two $k$-Fibonacci components

We study infinite paths of Markoff $m$-triples, that is, solutions to the generalised Markoff equation \[ x^2+y^2+z^2=3xyz+m, \] with $m>0$, with at least two $k$-Fibonacci components. First, we obtain a complete classification of Markoff $m$-triples whose last two entries are $k$-Fibonacci numbers and that are not roots of any Markoff trees. We then prove that every such infinite path is contained in a branch, starting at a triple of the form \[ \left(\frac{F_k(4r)}{3F_k(2r)},\,F_k(\ell+2r),\,F_k(\ell+4r)\right), \] where $r$ is an odd integer, $\ell\in\{1,2,\ldots, 2r\}$ and $3\nmid k$. These branches are distributed among exactly $2r$ distinct trees.

math.NT↗
arXiv2023

A binary quadratic approach to $X^2+(2k-1)^Y=k^Z$

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 structure of class groups of binary quadratic forms, we prove the conjecture when $4\Vert k$, with $2k-1$ a prime power and $4\le k\le 1000$.

math.NT↗
arXiv2019

A complete classification of well-rounded real quadratic ideal lattices

We provide a complete classification of well-rounded ideal lattices arising from real quadratic fields. We show that the ideals that give rise to such lattices are precisely the ones that correspond to divisors $a$ of the discriminant $d$ that satisfy $\sqrt{\frac{d}{3}}<a<\sqrt{3d}.$

math.NT↗
arXiv2017

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 $ε>0$, there is a positive integer $N$ such that if $α=2n\left(1+\dfrac{\log 2+ε}{\log n+j(n)}\right)$, then $R_n< p_{[α]}$ 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 $nj'(n)\to 0$.

math.NT↗
arXiv2013

On the prime divisors of elements of a $D(-1)$ 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.

math.NT↗