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Eteri Samsonadze

Publications and source records attributed to Eteri Samsonadze.

5 recordsLinked to original sources
arXiv2025

The sufficient conditions for insolvability of some Diophantine equations of $n$-th degree

The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number $c$ consists of powers of primes $p_i$ which satisfy the condition $φ(p_i^{k_i})\mid n$ ($p_i^{k_i }\geq 3)$ for some natural numbers $k_i$ $(i=1,2,\ldots ,l)$; $φ(x)$ is the Euler's totient function. Moreover, it is proved that if $b< m< p_i^{k_i}$ $(i=1,2,\ldots ,l)$, then this equation has no solution with natural components $x_1,x_2,\ldots ,x_m$. Besides, applying only elementary methods, it is proved that the Diophantine equation $x_1^n+x_2^n=(p^{s} p_1^{s_1} p_2^{s_2}\ldots p_l^{s_l})^{n}$ (with nonnegative integers $s$, $s_i$ $(i=1,2,..,l)$) has no solution with natural components if $n\geq 3$, $p$ is a prime number, while $p_i$ is a prime such that there is a natural number $k_i$ with $φ(p_i^{k_i})\mid n$ $(p_i^{k_i}\geq 3)$.

math.NT↗
arXiv2024

On sums of powers of natural numbers

The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved: $$\sum_{x=1}^{m}x^{n}=(-1)^{n}m(m+1)(-\frac{1}{2}+\sum_{i=2}^{n}a_i(m+2)(m+3)...(m+i)),$$ here $a_i=\frac{1}{i+1}\sum_{k=1}^{i}\frac{(-1)^{k}k^{n}}{k!(i-k)!}$, $(i=2,3,...,n-1)$, $a_n=\frac{(-1)^n}{n+1}$. Note that this formula does not contain Bernoulli numbers.

math.GM↗
arXiv2024

Sufficient conditions for solvability of linear Diophantine equations, and Frobenius numbers

The sufficient conditions for solvability of a linear Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ (with $a_1,a_2,...,a_n\in \mathbb{N}$) in non-negative integers $x_1,x_2,...,x_n$ are given. The explicit formulas are given for Frobenius numbers $g(a_1,a_2,...,a_n)$, for some particular cases,. Besides, a new recurrent method of studying the problem of solvability of a linear Diophantine equation in non-negative integers is proposed. This recurrent method is used for the problem of finding Frobenius numbers $g(a_1,a_2,...,a_n)$ for any $n\geq 3$; the example is given for the case $n=5$.

math.NT↗
arXiv2021

On the number of integer non-negative solutions of a linear Diophantine equation

We deal with the problem to find the number $P(b)$ of integer non-negative solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$ are natural numbers and $b$ is a non-negative integer. As different from the traditional methods of investigation of the function $P(b)$, in our study we do not employ the techniques of number series theory, but use in the main the properties of the Kronecker function and the elements of combinatorics. The formula is derived to express $P(b)$, for an integer non-negative $b$, via $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$ when $s\neq 0$, where $s=\left[ n-\dfrac{\sum_{i=1}^{n}a_i+r}{M} \right]$ and takes quite small values in some particular cases; $M$ is the least common multiple of the numbers $a_1,a_2,\ldots,a_n$, and $r$ is the remainder of $b$ modulo $M$. Also, the recurrent formulas are derived to calculate $P(b)$, for any non-negative integer $b$, which, in particular, are used in finding $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$. For the case where $s=0$ and $a_1$, $a_2$,..., $a_n$ are coprime, the explicit formula $P(b)=\dfrac{M^{n-1}}{a_1a_2\ldots a_n}C^{n-1}_{\left[\dfrac{b}{M}\right]+n-1}$ is given. To illustrate the proposed method, examples of finding the function $P(b)$ for linear Diophantine equations with $2,3,7$ and $n$ variables are given.

math.NT↗