Download On three consecutive primes

On three consecutive primes
Tsutomu Hashimoto
September, 2009
Abstract
In this paper, we prove certain theorems about three consecutive primes.
Introduction
The following result is due to Ishikawa [5].
Theorem 1.
If
is the
th prime, then
for all
.
We prove stronger results.
Theorem 2.
If
4, the inequality
holds.
Theorem 3.
If
9, the inequality
holds.
Theorem 4.
Let
and
be any real numbers with
0. Then
2
for all
1.
some
,
2
.
Proof of Theorem 1
According to "A Theorem of Sylvester and Schur" [2] (see also [1]), there is a number
containing a prime divisor
in the sequence
. It
2, … ,
in this sequence, that
follows, since there's not a multiple of
2.
1,
. Proof of Theorem 2
According to Nagura [4] (see also [3]),
1
1⁄ 5
for
9.
Hence by a calculation
1⁄ 5
2460
229
2400
By actual verification, we find that it is true for smaller values. 1
0 9.
3.
Proof of Theorem 3
According to Rohrbach & Weis [6] (see also [3]),
1⁄13
1
for
118.
Hence by a calculation
1⁄13
23658180
3325841
56847882
38416742
0 118.
By actual verification, we find that it is true for smaller values. 4.
Proof of Theorem 4
follows,
2
since
some
2
because
⁄
lim
lim
ε for all
⁄
⁄
stands for the sequence
Suppose that the notation
,
1
1
. It
2 ,
that
. Hence
ε
⁄
⁄
⁄
2
2
ε
,
ε. Acknowledgement
I thank Jonathan Sondow for pointing out Ishikawa's paper and for suggesting Theorems 3 and 4.
References
[1] Sondow, Jonathan and Weisstein, Eric W. "Bertrand's Postulate." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/BertrandsPostulate.html.
[2] Erdős, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934.
[3] Caldwell, C. "The Gaps Between Primes." http://primes.utm.edu/notes/gaps.html.
[4] Nagura, J. "On the interval containing at least one prime number." Proceedings of the Japan
Academy, Series A 28 (1952), pp. 177--181.
[5] Ishikawa, H. "Über die Verteilung der Primzahlen." Science Rep. Tokyo Bunrika Daigaku 2, 27-4
(1934).
[6] Hans Rohrbach and Jürgen Weis, "Zum finiten Fall des Bertrandschen Postulats", J. Reine Angew.
Math., 214/215 (1964), 432–440.
Shiga 520-2412 JAPAN
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