Affiliation:
1. Department of Mathematics and Statistics , University of Turku , 20014 Turku , Finland
2. School of Applied Mathematics , Nanjing University of Finance and Economics , Nanjing 210046 , P. R. China
Abstract
Abstract
Let
p
n
p_{n}
denote the 𝑛-th prime.
We prove that, for any
m
≥
1
m\geq 1
, there exist infinitely many 𝑛 such that
p
n
-
p
n
-
m
≤
C
m
p_{n}-p_{n-m}\leq C_{m}
for some large constant
C
m
>
0
C_{m}>0
, and
p
n
+
1
-
p
n
≥
c
m
log
n
log
log
n
log
log
log
log
n
log
log
log
n
p_{n+1}-p_{n}\geq\frac{c_{m}\log n\log\log n\log\log\log\log n}{\log\log\log n}
for some small constant
c
m
>
0
c_{m}>0
.
Furthermore, for any fixed positive integer 𝑙, there are many positive integers 𝑘 with
(
k
,
l
)
=
1
(k,l)=1
such that
p
′
(
k
,
l
)
≥
c
k
⋅
log
k
log
log
k
log
log
log
log
k
log
log
log
k
,
p^{\prime}(k,l)\geq ck\cdot\frac{\log k\log\log k\log\log\log\log k}{\log\log\log k},
where
p
′
(
k
,
l
)
p^{\prime}(k,l)
denotes the least prime of the form
k
n
+
l
kn+l
with
n
≥
1
n\geq 1
, which improves the previous result of Prachar.
Subject
Applied Mathematics,General Mathematics