Affiliation:
1. Faculty of Applied Mathematics and Informatics , Technical University of Sofia , 8, St. Kliment Ohridski Blvd., 1756 Sofia , Bulgaria
Abstract
Abstract
In this paper, we introduce a new diophantine equation with prime numbers.
Let
[
⋅
]
[\,\cdot\,]
be the floor function.
We prove that, when
1
<
c
<
23
21
1<c<\frac{23}{21}
and
θ
>
1
\theta>1
is fixed, then every sufficiently large positive integer 𝑁 can be represented in the form
N
=
[
p
1
c
tan
θ
(
log
p
1
)
]
+
[
p
2
c
tan
θ
(
log
p
2
)
]
+
[
p
3
c
tan
θ
(
log
p
3
)
]
,
N=[p^{c}_{1}\tan^{\theta}(\log p_{1})]+[p^{c}_{2}\tan^{\theta}(\log p_{2})]+[p^{c}_{3}\tan^{\theta}(\log p_{3})],
where
p
1
,
p
2
,
p
3
p_{1},p_{2},p_{3}
are prime numbers.
We also establish an asymptotic formula for the number of such representations.
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