Re-extending Chebyshev’s theorem about Bertrand’s conjecture
In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal...
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| author | Shams, Armіn Шамс, Армін |
| author_facet | Shams, Armіn Шамс, Армін |
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| description | In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal gaps table is used to verify initial states. The extended theorem contains a constant r, which can be reduced if more initial states can be checked. Therefore, the theorem can be even more extended when maximal gaps table is extended. The main extension idea is not based on r, though. |
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K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDC 517.93
Armin Shams (Univ. Manchester, UK; Ferdowsi Univ. Mashhad, Iran)
RE-EXTENDING CHEBYSHEV’S THEOREM
ABOUT BERTRAND’S CONJECTURE
POVTORNE ROZÍYRENNQ TEOREMY ÇEBYÍOVA
WODO HIPOTEZY BERTRANA
In this paper, Chebyshev's theorem (1850) about Bertrand’s conjecture is re-extended using a theorem
about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this
extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal
gaps table is used to verify initial states. The extended theorem contains a constant, r, which can be
reduced if more initial states can be checked. Therefore, the theorem can be even more extended when
maximal gaps table is extended. The main extension idea is not based on r, though.
U danij statti teoremu Çebyßova (1850) wodo hipotezy Bertrana povtorno rozßyreno za dopo-
mohog teoremy wodo hipotezy Serpins\koho (1958). Raniße teoremu rozßyrgvaly dekil\ka
raziv, ale rozhlqduvane rozßyrennq [ najholovnißym iz poperednix. Dovedennq poçyna[t\sq z
vykorystannq tablyci maksymal\nyx promiΩkiv dlq perevirky poçatkovyx staniv. Rozßyrena
teorema mistyt\ konstantu r, qka moΩe buty zmenßena pry moΩlyvosti perevirky bil\ßo]
kil\kosti poçatkovyx staniv. OtΩe, teoremu moΩe buty rozßyreno navit\ bil\ße u vypadku roz-
ßyrennq tablyci maksymal\nyx promiΩkiv. Prote osnovna ideq rozßyrennq ne bazu[t\sq na r.
1. Introduction. We are going go introduce and prove the following two theorems
(mainly the first one):
Theorem 1. For each real number m not less than 2, there exists at least one
prime number p such that:
m < p < m + 2 × m
mrelog
, r = 1.207.
Theorem 2. Each natural number n can be written as sum of distinct prime-
based squares and/or base-2 powers of distinct prime numbers or 1:
∀ n ∈ N ∃ a1, a2, … , ax , b1, b2 , … , by ∈ P ∪ {1}:
∀ 1 ≤ k, l ≤ x a ak l≠ , ∀ 1 ≤ k, l ≤ y b bk l≠ ,
n a
i
x
i
j
y
bj= +
= =
∑ ∑
1
2
1
2 ,
P is the set of prime numbers and N is the set of natural numbers.
© ARMIN SHAMS, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12 1701
1702 ARMIN SHAMS
In our proofs, we somehow deal with one conjecture and two theorems:
Conjecture 1 (Sierpinski, 1958). If we form an N × N (N ∈ N – {1}) square
using consecutive natural numbers from 1 to N2, then there exists at least one
prime number in each row of this square (the square is called Sierpinski’s square)
[1, p. 732].
Theorem 3. For each n ek> (n, k ∈ N), there exists at least one prime number
in each first k row of Sierpinski’s square [1, p. 732].
Theorem 4 (Chebyshev, 1850). For each natural number n, there exists at least
one prime number p such that:
n < p < 2n (Erdos (1932): n < p < 2n – 2, n > 3) [1, p. 791].
Note that this theorem was extended at least three times after 1932; but extensions
were made such that they were true for numbers upper that a specific non-small
number:
∀ n ∈ N ∃ p : n < p < 1 1
5
+
n for n ≥ P10 (Nagura, 1952) [2],
∀ n ∈ N ∃ p : n < p < 1 1
13
+
n for n ≥ P119 (Rohrbach & Weis, 1964) [2],
∀ n ∈ N ∃ p : n < p < 1 1
16597
+
n for n ≥ P2010761 (Schoenfeld, 1976) [2].
Above theorems are better than our suggested re-extended theorem for some limited
initial states, but after those states, they are weaker than ours for all other unlimited
remaining states.
2. Extending the Theorem 4. The Theorem 4 can be extended as follows:
For first 1693182318746371 numbers computer can check the theorem, but it
takes a very long time if an ordinary computer is used. If we consider the table of
maximal gaps between primes not greater than 1693182318746371 [3], obtained by
super computers, then we can soon realize that the theorem is acceptable for initial
numbers up to 1693182318746371.
If we assume r = 1.207 and m ≥ 1693182318746371 (m is a positive integer),
then we can write m ≥ ( )re 29 and thus logre m ≥ 29. On the other hand, first
derivative of 1 207. x – 8 × x is positive for x ≥ 29. Thus 1 207. x – 8 × x is an
increasing function for x ≥ 29 and we know that this function is positive for x = 29.
Therefore 1 207. x – 8 × x > 0, for x ≥ 29, that concludes 8 × x < 1 207. x , x ≥ 29. We
can substitute 1.207 by r, and x by logre m. We will have
log log
re
mm r re× <8 . (1)
Reversing both sides and multiplying them by m we can claim that
m
r
m
mre m
re
log log ( )
<
× 8
. (2)
If we replace m with ( )logre re m in left side, we will have
( )
log ( )
log
log
re
r
m
m
re
re
m
m
re
<
× 8
. (3)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
RE-EXTENDING CHEBYSHEV’S THEOREM ABOUT BERTRAND’S CONJECTURE 1703
If we take natural logarithm from both sides, we will have
log ln
log
.re
re
m m
m
<
− 2 07. (4)
So we can write
m m
m
m
mre re
<
−
log
ln
log
.2 07 . (5)
According to m ≥ 1693182318746371, we can write ln
log
m
mre
– ln
log
m
mre
<
< 0.01 (to prove this we can raise e to the power of both sides and rewrite the sides
according to x = m
mrelog
> e30 and [x] ≥ x – 1 and x
x − 1
= 1 + 1
1x −
and e0 01. >
> 1.005) .
And then we can write
m m
m
m
m
m
m
m
mre re re re
<
− + −
−
log
ln
log
. . ln
log
ln
log
2 07 0 01 . (6)
Obviously
m m
m
m
mre re
<
−
log
ln
log
.2 06 . (7)
If x ≥ en , n ≥ 1, then ln x < x
n
, therefore according to m ≥ 1693182318746371,
which means m
mrelog
> e30 we can write ln
log
m
mre
< 0.06 × m
mrelog
; therefore
ln
log
m
mre
– 2 < 0.06 × m
mrelog
; so
m m
m
m
m
m
m
m
mre re re re
<
−
+ × −
−
log
ln
log
. .
log
ln
log
2 06 0 06 2 . (8)
We will have
m m
m
m
mre re
< −
−
log
ln
log
1 2 . (9)
We can write
m m
m
m
mre re
<
−
log
ln
log
2 . (10)
Assuming n = m
mrelog
we conclude m < n (ln n – 2). According to [ln n] ≥ ln n – 1
we will have m < n ( [ ln n ] – 1) . Using the Theorem 3 we can write ∃ k : k = [ ln n ].
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1704 ARMIN SHAMS
So we conclude
m < n ( k – 1) . (11)
Using the found n, m is somewhere in the first k – 1 rows of the Sierpinski’s square
where there is at least one prime number in each row. The worst case takes place when
m is prime and the next prime is in the last column of the next row. So we can write
∀ m ∈ N ∃ p : m < p < m + 2n, m ≥ 2. (12)
So
∀ m ∈ R ∃ p : [m] < p < [m] + 2 × [ ]
[ ]
m
mrelog
, m ≥ 2, (13)
p is natural, m – [m] is less than 1 and by taking derivative we can prove that
m
mrelog
and thus f m( ) = m + 2 × m
mrelog
are increasing functions for m ≥ re, so,
because m ≥ [m], f m( ) is not less than f m[ ]( ) = [m] + 2 × [ ]
[ ]
m
mrelog
, m ≥ re.
Therefore we have
∀ m ∈ R ∃ p : [m] + m m− [ ]( ) < p < m + 2 × m
mrelog
, m ≥ re. (14)
So
∀ m ∈ R ∃ p : m < p < m + 2 × m
mrelog
, m ≥ re. (15)
Therefore, based on 2 × m
mrelog
> 2 if m > 1, the theorem can be extended as:
The extended theorem. For each real number m > 1, there exists at least one
prime number p such that:
m < p < m + 2 × m
mrelog
, r = 1.207.
Important note. We can reduce r if we satisfy part (1) with a large m.
According to the computational power of the computers which have been used to
produce the maximal gaps table (in Appendix), it is provable that r = 1.207 (or “r”
greater than 1.207 that we do not need).
3. The new theorem (Theorem 2, provable using the extended theorem). We
will use following lemma to prove Theorem 2:
Lemma. For each n ≥ 121 there exists p such that n
2
< p2 < n.
Proof. It is sufficient to check a few initial states, and then use the Nagura
theorem (page 2) to obtain ∀ m ≥ 11 ∃ p : m < p < 2 m. For each natural n not
less than 121, if we assume m = n
2
, then n
2
≥ 11 and n
2
∈ R, therefore there
exists p such that
n
2
< p < 2 n
2
. So: n
2
< p2 < n, n ≥ 121.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
RE-EXTENDING CHEBYSHEV’S THEOREM ABOUT BERTRAND’S CONJECTURE 1705
Theorem 5. Each natural number n can be written as sum of distinct prime-
based squares and/or base-2 powers of distinct prime numbers or 1:
∀ n ∈ N ∃ a1, a2, … , ax , b1, b2 , … , by ∈ P ∪ {1}:
∀ 1 ≤ k, l ≤ x a ak l≠ , ∀ 1 ≤ k, l ≤ y b bk l≠ ,
n a
i
x
i
j
y
bj= +
= =
∑ ∑
1
2
1
2 .
Besides, it can be shown that bj can be kept less than or equal to 5 and thus y
can be kept less than or equal to 4.
Proof. For n less than 121, the theorem can be easily verified. To use strong
induction, it’s enough to prove that if the theorem is true for 1 to n, then it is also
true for n + 1. According to the lemma, we can find p such that
n + 1
2
< p2 < n + 1.
It is sufficient to assume that, p is one of “ai”s and obtain other “ ai”s from n + 1 –
– p2 for which, the truth of theorem is previously proved according to induction
assumption. Obviously based on induction assumption, obtained “ ai”s from n + 1 –
– p2 are different (as well as “ bi ”s) and p is different with all “ ai”s it
follows
from the relation n + 1 = p2 + q that 0 ≤ q ≤
n + 1
2
≤ p2 , and hence, the
representation q = ′=∑ aii
x 2
1
+
j
y bj
=
′∑ 1
2 contains ′ai < p for every i = 1, … , x) .
Therefore out theorem is true for n + 1; so, based on strong induction, the theorem is
true for all natural n.
4. Conclusions. The extended theorem is a fundamental theorem in prime number
theory. Based on this extension, several other extensions can be made to related
theorems.
In the extended theorem, if the truth of the theorem is proved for all m < C then
the coefficient r can be reduced to approach 1+. C depends on r and is a very large
number if r approaches 1. C can be obtained from part (1) of the proof and is the
least m that satisfies the “<” sign according to r. The number “8” in part (1) can be
a little reduced currently by minor changes to the proof.
For large C (but not very large), the truth of the theorem for all m < C can be
checked by super computer. Reduction of r is limited by the computational power of
the computer and the efficiency of the algorithm. By using more efficient programs or
more powerful computers we can check initial states for a greater C (for all m < C)
and reduce r even more, approaching r = 1.
One important result of the article is that if Sierpinski’s conjecture is proved, the
theorem can be considerably re-extended by a proof method similar to the one
discussed in the article.
5. Acknowledgments. The author thanks A. M. Naranjani and Ramtin Shams for
their valuable suggestions. The author did the original work in the warm scientific
community of the Computer Engineering Department of Ferdowsi University of
Mashhad in 2001.
Appendix. In the following table we list the maximal gaps through 1131. These
are the first occurrences of gaps of at least of this length. For example, there is a gap of
879 composites after the prime 277900416100927. This is the first occurrence of a
gap of this length, but still is not a maximal gap since 905 composites follow the
smaller prime 218209405436543 [5].
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1706 ARMIN SHAMS
Gap Following the prime Reference
0 2
1 3
3 7
5 23
7 89
13 113
17 523
19 887
. .
. .
. .
581 1346294310749
587 1408695493609
601 1968188556461
651 2614941710599
673 7177162611713
715 13829048559701 [4]
765 19581334192423 [4]
777 42842283925351 [4]
803 90874329411493 [5]
805 171231342420521 [5]
905 21820940543643 [5]
915 1189459969825483 [6]
923 1686994940955803 [6]
1131 1693182318746371 [6]
1. Mosaheb Gh. Elementary number theory. – Tehran: Soroush Press, 1980. – Vol. 2, pt 2.
2. Ribenboim P. The new book of prime number records. – 3 rd Edition. – New York: Springer, 1995.
3. Caldwell Ch. K. http://www.utm.edu/research/primes/notes/GapsTable.html.
4. Young J., Potler A. First occurrence of prime gaps // Math. Comput. – 1989. – 53. – P. 221 – 224.
5. Nicely T. New maximal prime gaps and first occurrence // Ibid. – 1999. – 68. – P. 1311 – 1315.
6. Nicely T., Nyman B. First occurrence of a prime gap of 1000 or greater (preprint available at
http://www.trnicely.net/index.html).
Received 13.06.06,
after revision — 18.01.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
|
| id | umjimathkievua-article-3424 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:18Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0b/ca647e76f10a6531678400eee680dc0b.pdf |
| spelling | umjimathkievua-article-34242020-03-18T19:53:47Z Re-extending Chebyshev’s theorem about Bertrand’s conjecture повторне розширення теореми Чебишова щодо гіпотези Бертрана Shams, Armіn Шамс, Армін In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal gaps table is used to verify initial states. The extended theorem contains a constant r, which can be reduced if more initial states can be checked. Therefore, the theorem can be even more extended when maximal gaps table is extended. The main extension idea is not based on r, though. У даній статті теорему Чебишова (1850) щодо гіпотези Вертрана повторно розширено за допомогою теореми щодо гіпотези Серпінського (1958). Раніше теорему розширювали декілька разів, але розглядуване розширення є найголовнішим із попередніх. Доведення починається з використання таблиці максимальних проміжків для перевірки початкових станів. Розширена теорема містить константу r, яка може бути зменшена при можливості перевірки більшої кількості початкових станів. Отже, теорему може бути розширено навіть більше у випадку розширення таблиці максимальних проміжків. Проте основна ідея розширення не базується на r. Institute of Mathematics, NAS of Ukraine 2007-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3424 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 12 (2007); 1701–1706 Український математичний журнал; Том 59 № 12 (2007); 1701–1706 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3424/3590 https://umj.imath.kiev.ua/index.php/umj/article/view/3424/3591 Copyright (c) 2007 Shams Armіn |
| spellingShingle | Shams, Armіn Шамс, Армін Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title | Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title_alt | повторне розширення теореми Чебишова щодо гіпотези Бертрана |
| title_full | Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title_fullStr | Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title_full_unstemmed | Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title_short | Re-extending Chebyshev’s theorem about Bertrand’s conjecture |
| title_sort | re-extending chebyshev’s theorem about bertrand’s conjecture |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3424 |
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