There always exists at least one prime number
between x
2
and x
2
+ x
Wan-Dong Xu
School of Science, Tianjin University, Tianjin, 300072, China
e-mail: xwandong@sohu.com
Abstract
In this pap er, one has shown that there always exists at least one pseudo
prime number between x
2
and x
2
+ x in a pseudo sequence of odd
numbers, so it also is true in the real sequence of odd numbers.
Keywords: Prime, distribution of primes
1. Introduction
In earlier paper [1], one has shown the Desbove conjecture: there always
exists at least one prime between x
2
and (x + 1)
2
in {O
x
} for every integer
x ≥ 1. There is another conjecture: there always exists at least one prime
between x
2
and x
2
+ x for every integer x > 1. It is a stranger conjecture
owing to that the interval between x
2
and x
2
+ x is a half of that between
x
2
and (x + 1)
2
. One suspects it possible that there would be not any prime
in anyone of those smaller intervals. It is important for understanding the
distribution of prime in sequence of natural number to show whether exist
primes in the intervals, so that this problem was elected as eleventh one in
distribution field of prime by American mathematician T. M. Apostol [2].
In this paper one will try to solve this problem with the method analogous
as the earlier paper, and one will use same mathematical symbols as that
paper.
2. There always exists at least one pseudo prime between x
2
and x
2
+ x in {P
x
} for every integer x > 1
In order to research the number of prime numbers in the intervals between
x
2
and x
2
+ x for the sequence {O
x
} and that of pseudo prime numbers in
the same intervals for the sequence {P
x
}, one redivides both sequences {O
x
}
and {P
x
} into infinitely many of intervals respectively.
1
http://www.paper.edu.cn
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