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Sagar Mandal

Publications and source records attributed to Sagar Mandal.

13 recordsLinked to original sources
arXiv2026

On common values of $F_n$ and Nathanson's totient function $\Phi(m)$

In a recent paper, Chatterjee, the author and Mohan posed the problem of determining all solutions of the Diophantine equation $F_n=\Phi(m)$, where $F_n$ is the $n$-th Fibonacci number and $\Phi(m)$ counts the number of nonempty sets $A \subseteq \{1, 2, \dots, m\}$ for which $\gcd(A)$ is relatively prime to $m$. In this paper, we prove that the Diophantine equation has the only solutions $(n,m)=(1,1),(2,1),(3,2)$. The main tools used in this paper are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth{\H{o}} version of the Baker-Davenport reduction method in diophantine approximation.

math.GM↗
arXiv2026

Exact and Asymptotic Counts of MSTD, MDTS, and Balanced Sets in Dicyclic Groups

We investigate the relationship between the sizes of the sum and difference sets of the Dicyclic Group $\mathrm{Dic}_{4n}$. We first determine the exact numbers of MSTD (more sums than differences), MDTS (more differences than sums), and balanced subsets of size two. As a consequence, we show that the numbers of MSTD and balanced subsets of size two are asymptotically equal as $n \to \infty$. For odd $n$, we then obtain exact counts of MSTD, MDTS, and balanced subsets of size three, with the results depending on whether $n$ is divisible by $3$. In this case, we establish that asymptotically the number of MSTD subsets of size three is six times the number of MDTS subsets and also six times the number of balanced subsets. Finally, we establish a lower bound for the number of MSTD, MDTS, and balanced subsets of $\mathrm{Dic}_{4n}$ corresponding to the boundary case of size $2n$.

math.GM↗
arXiv2026

On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions

The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $\tau(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-periodic. Further, we deduce that the sequence arising from Nathanson's totient function $\Phi(n)$, the classical Euler's totient function $\varphi(n)$, sum of divisor function $\sigma(n)$, their Dirichlet convolution $\sigma*\varphi(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $\varphi^*(n)$ modulo $m$, are non-periodic for certain modulo $m$. In addition, we extend a result of Ayad and Kihel \cite{r1} on the non-periodicity of certain arithmetic function $g(n)$. On the other hand, we construct several transcendental numbers arising from the continued fractions attached with $\tau(n)$, $E_k(z)$, $\Phi(n)$, $g(n)$, $\varphi(n)$, $\sigma(n)$, $\sigma*\varphi(n)$, $J_k(n)$, and $\varphi^*(n)$.

math.GM↗
arXiv2025

Divisibility and Sequence Properties of $\sigma^+$ and $\varphi^+$

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $\varphi(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient function $\varphi^{*}(n)$, we investigate analogous divisibility problems involving the functions $\sigma(n)$, $\sigma^{+}(n)$, and $\varphi^{+}(n)$. Further, we establish some interesting properties of the sequences $\left\{\sigma^+(n)\right\}_{n=1}^\infty$ and $\left\{\varphi^+(n)\right\}_{n=1}^\infty$, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$.

math.GM↗
arXiv2025

A Note on Deaconescu's Conjecture

Hasanalizade [1] studied Deaconescu's conjecture for positive composite integer $n$. A positive composite integer $n\geq4$ is said to be a Deaconescu number if $S_2(n)\mid \phi(n)-1$. In this paper, we improve Hasanalizade's result by proving that a Deaconescu number $n$ must have at least seventeen distinct prime divisors, i.e., $\omega(n)\geq 17$ and must be strictly larger than $5.86\cdot10^{22}$. Further, we prove that if any Deaconescu number $n$ has all prime divisors greater than or equal to $11$, then $\omega(n)\geq p^{*}$, where $p^{*}$ is the smallest prime divisor of $n$ and if $n\in D_3$ then all the prime divisors of $n$ must be congruent to $2$ modulo $3$ and $\omega(n)\geq 48$.

math.GM↗
arXiv2025

Exploring the Relationships Between the Divisors of Friends of $10$

A solitary number is a positive integer that shares its abundancy index only with itself. $10$ is the smallest positive integer suspected to be solitary, but no proof has been established so far. In this paper, we prove that not all half of the exponents of the prime divisors of a friend of 10 are congruent to $1$ modulo $3$. Furthermore, we prove that if $F=5^{2a}\cdot Q^2$ ($Q$ is an odd positive integer coprime to $15$) is a friend of $10$, then $\sigma(5^{2a})+\sigma(Q^2)$ is congruent to $6$ modulo $8$ if and only if $a$ is even, and $\sigma(5^{2a}) + \sigma(Q^2)$ is congruent to $2$ modulo $8$ if and only if $a$ is odd. In addition, if we set $Q={\displaystyle \prod_{i=2}^{\omega(F)}}p_{i}^{a_i}$ and $a=a_1$, where $p_i$ are prime numbers, then we establish that $$F>\frac{25}{81}\cdot\prod_{i=1}^{\omega(F)}(2a_i + 1)^2,$$ in particular $F> 625\cdot 9^{\omega(F)-3}.$

math.GM↗
arXiv2025

A note on solitary numbers

Does $14$ have a friend? Until now, this has been an open question. In this note, we prove that a potential friend $F$ of $14$ is an odd, non-square positive integer. $7$ appears in the prime factorization of $F$ with an even exponent while at most two prime divisors of $F$ can have odd exponents in the prime factorization of $F$. If $p\mid F$ such that $p$ is congruent to $7$ modulo $8$, then $p^{2a}\mid\mid F$, for some positive integer $a$. Further, no prime divisor of $F$ has an exponent congruent to $7$ modulo $8$ and no prime divisor can exceed $1.4\sqrt{F}$. The primes $3,5$ cannot appear simultaneously in the prime factorization of $F$. If $(3,F)>1$ or $(5,F)>1$, then $\omega(F)\geq4$, otherwise $\omega(F)\geq8$.

math.GM↗
arXiv2024

Prime Divisors of 10's Friends: A Generalization of Prior Bounds

10 is the smallest positive integer which is whether solitary or friendly is still an open question in mathematics. In this paper, we provide upper bounds for each of the prime divisors of a friend of 10. This paper is precisely a generalization of a recent paper [4] in which necessary upper bounds for the 2nd, 3rd, and 4th smallest prime divisors of a friend of 10 have been proved. Further, we establish better upper bounds for the 3rd, and 4th smallest prime divisors of a friend of 10 than the bounds given in [4].

math.GM↗
arXiv2024

On Characterizing Potential Friends of 20

Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-5$ and if $\Omega(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $\Omega(n)$ and $\omega(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$.

math.GM↗
arXiv2024

A note on necessary conditions for a friend of 10

Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ where $f$ is the smallest odd positive integer greater than $1$ and less than or equal to $\min\{ 2a+1,p-1\}$, provided $5^{2a}\mid \mid N$. Further, there exist prime factors $p$ and $q$ (not necessarily distinct) of $N$ such that $p\equiv1 \pmod {10}$ and $q\equiv 1\pmod 6$. Besides, we prove that if a Fermat prime $F_k$ divides $N$ then $N$ must have a prime factor congruent to $1$ modulo $2F_k$. Also, if we consider the form of $N$ as $N=5^{2a}m^2$ then $m$ is non square-free. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-4$ and if $\Omega(m)\leq K$ then $N< 5\cdot 6^{(2^{K-2a+1}-1)^2}$ where $\Omega(n)$ and $\omega(n)$ denote the total number of prime factors and the number of distinct prime factors of the integer $n$ respectively.

math.NT↗
arXiv2021

Nanostructured Plasmonic Metal Surfaces as Optical Components for Infrared Imaging and Sensing

Thermal imaging and sensing technologies offer critical information about our thermally radiant world, and in recent years, have seen dramatic increases in usage for a range of applications. However, the cost and technical finesse of manufacturing infrared optical components remain a major barrier towards the democratization of these technologies. In this report, we present a solution processed plasmonic reflective filter or PRF as a scalable and inexpensive thermal infrared optic. The PRF selectively absorbs sunlight and specularly reflects thermal infrared TIR wavelengths with performance comparable to state-of-the-art TIR optics made of materials like Germanium. Unlike traditional infrared optical components, however, the PRF can be conveniently fabricated using inexpensive materials and a dip and dry chemical synthesis technique, and crucially, has manufacturing costs that are orders of magnitude lower. We experimentally demonstrate the core optical functionality of the PRF, as well as its integration into infrared imaging and sensing systems without compromising their thermographic or radiometric capabilities. From a practical standpoint, the inexpensive and convenient fabricability of the PRF represent a significant advance towards making the benefits of thermal imaging and sensing systems more affordable and accessible. Scientifically, our work demonstrates a previously unexplored optical functionality and a new direction for versatile chemical synthesis in designing optical components.

physics.optics↗
arXiv2021

Radiative Cooling and Thermoregulation in the Earth's Glow

Passive radiative cooling involves a net radiative heat loss into the cold outer space through the atmospheric transmission windows. Due to its passive nature and net cooling effect, it is a promising alternative or complement to electrical cooling. For efficient radiative cooling of objects, an unimpeded view of the sky is ideal. However, the view of the sky is usually limited - for instance, the walls of buildings have >50% of their field of view subtended by the earth. Moreover, objects on earth become sources of heat under sunlight. Therefore, building walls with hot terrestrial objects in view experience reduced cooling or heating, even with materials optimized for heat loss into the sky. We show that by using materials with selective long-wavelength infrared (LWIR) emittances, vertical building facades experience higher cooling than achievable by using broadband thermal emitters like typical building envelopes. Intriguingly, this effect is pronounced in the summer and diminishes or even reverses during the winter, indicating a thermoregulation effect. The findings highlight a major opportunity to harness untapped energy savings in buildings.

physics.app-ph↗