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

Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let \xi(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this…

We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a…

Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in…

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime…

When $k\ge 4$ and $0\le d\le (k-2)/4$, we consider the system of Diophantine equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne k-d). \] We show that in this cousin of a Vinogradov system, there is a paucity of…

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive…

We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1 < c < \frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the…

For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative…

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$…

We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to…

We refine a result of W.P. Li and Wang on the values of the form $ \lambda_1p_1 + \lambda_2p_2^{2} + \lambda_3p_3^{2} + \mu_1 2^{m_1} +...+ \mu_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers,…

Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of…

In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon…

In [1] it is shown that the Diophantine equation $(k!)^n+k^n=(n!)^k+n^k$ only has the trivial solution $n=k$, and $(k!)^n-k^n=(n!)^k-n^k$ only has the solutions $n=k$, $(n, k)=(1, 2),$ and $(2, 1)$. In this article we find all solutions of…

Motivated by the recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized…