dorsal/arxiv
View SchemaA Diophantine inequality involving different powers of primes of the form {\boldmath$[n^c]$}
| Authors | S. I. Dimitrov |
|---|---|
| Categories | |
| ArXiv ID | 2601.09405vv1 |
| URL | https://arxiv.org/abs/2601.09405 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let $\frac{219}{220}<\gamma<1$ and $\theta>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda_1p_1 + \lambda_2p_2 + \lambda_3p^4_3+\eta|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220\gamma}{208}+\theta} \end{equation*} and such that $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3$.
{
"annotation_id": "8ad2ae33-70b0-476a-9295-d3972e03c50d",
"date_created": "2026-02-17T05:53:19.901000Z",
"date_modified": "2026-02-17T05:53:19.901000Z",
"file_hash": "dfbbd1f487c2cbf13a45abc936ccb03c5caf3a9ce9d1a7c56fa65b99a6ee2d1a",
"private": false,
"record": {
"abstract": "Let $[\\, x\\,]$ denote the integer part of a real number $x$. Assume that $\\lambda_1,\\lambda_2,\\lambda_3$ are nonzero real numbers, not all of the same sign, that $\\lambda_1/\\lambda_2$ is irrational, and that $\\eta$ is real. Let $\\frac{219}{220}\u003c\\gamma\u003c1$ and $\\theta\u003e0$. We establish that, there exist infinitely many triples of primes $p_1,\\, p_2,\\, p_3$ satisfying the inequality \\begin{equation*} |\\lambda_1p_1 + \\lambda_2p_2 + \\lambda_3p^4_3+\\eta|\u003c\\big(\\max \\{p_1, p_2, p^4_3\\}\\big)^{\\frac{219-220\\gamma}{208}+\\theta} \\end{equation*} and such that $p_i=[n_i^{1/\\gamma}]$, $i=1,\\,2,\\,3$.",
"arxiv_id": "2601.09405",
"authors": [
"S. I. Dimitrov"
],
"categories": [
"math.NT"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "A Diophantine inequality involving different powers of primes of the form {\\boldmath$[n^c]$}",
"url": "https://arxiv.org/abs/2601.09405",
"version": "v1"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "777e848e-10e7-4cdc-995d-9264b4d8605e",
"id": "arXiv Dataset",
"type": "Model",
"variant": "snapshot-2026-01-17",
"version": "0.1.0"
},
"user_id": 1000002
}