| Issue |
RAIRO-Theor. Inf. Appl.
Volume 60, 2026
Diophantine Analysis and Related Topics (DART2025Z)
|
|
|---|---|---|
| Article Number | 4 | |
| Number of page(s) | 21 | |
| DOI | https://doi.org/10.1051/ita/2026002 | |
| Published online | 25 February 2026 | |
A Diophantine inequality with one prime of the form p = m2 + n2 + 1
School of Mathematics and Statistics, Shandong Normal University,
Jinan
250014,
Shandong,
PR China
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
4
November
2025
Accepted:
31
December
2025
Abstract
Let 1 < c < (300s + 31)/(200s + 64) be fixed. Assume that N > 0 is a large enough number and ε > 0 is an arbitrarily small constant. This paper establish that the Diophantine inequality
|p1c + … + psc − N | < ε
has a solution in prime numbers p1,…,ps, where s ≥ 7 and s is a natural number and p1 can be expressed as p1 = m2 + n2 + 1 for some integers m,n.
Mathematics Subject Classification: 11D75 / 11L07 / 11P32
Key words: Diophantine inequality / Exponential sum / Primes / Bombieri–Vinogradov type result
© The authors. Published by EDP Sciences, 2026
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