Related papers: Waring problem with the Ramanujan $\tau$-function

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $\tau$ function applied to $n$.

We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we…

We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[…

We shall show that, for any positive integer $D>0$ and any primes $p_1, p_2$ not dividing $D$, the diophantine equation $x^2+D=2^s p_1^k p_2^l$ has at most $63$ integer solutions $(x, k, l, s)$ with $x, k, l\geq 0$ and $s\in \{0, 2\}$.

Let \tau(.) be the Ramanujan \tau-function, and let k be a positive integer such that \tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set…

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

Ramanujan showed that $\tau(p) \equiv p^{11}+1 \pmod{691}$, where $\tau(n)$ is the $n$-th Fourier coefficient of the unique normalized cusp form of weight $12$ and full level, and the prime $691$ appears in the numerator of…

Let $\tau$ denote the Ramanujan tau function. One is interested in possible prime values of $\tau$ function. Since $\tau$ is multiplicative and $\tau(n)$ is odd if and only if $n$ is an odd square, we only need to consider $\tau(p^{2n})$…

We study the prime values of Ramanujan's tau function $\tau(n)$. Lehmer found that $n=251^2=63001$ is the smallest $n$ such that $\tau(n)$ is prime: $$\tau(251^2)=-80561663527802406257321747.$$ We prove that in most arithmetic progressions…

Ramanujan found five solutions to the exponential Diophantine equation $z^2+7= 2^n$ where $z$ and $n$ are positive integers, and posed the problem of determining whether there are any more. Nagell was the first to prove that there were not.…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

In this study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.

We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}$. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for…

Let $1<c<\frac{1787}{1502}$ and $N$ be a sufficiently large real number. In this paper, it is proved that for any arbitrarily large number $E>0$ and for almost all real $R \in (N,2N]$, the Diophantine inequality…

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous…

Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $\tau(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.

Using the simple properties of Riemman integrable functions, Ramanujan's formula for sum of the square roots of first n natural numbers has been generalized to include r'th roots where r is any real number greater than 1.As an application…