CHAPTER 2

On the history of forming prime numbers tables and determining the smallest divisor of composite numbers

 

 

2.1. 2.1. Famous tables of prime numbers and divisors of composite numbers

Considering the importance of recognition the prime numbers and factorization numbers to them and in other side, difficulties of recognition of primality and factorization, some tables (of course restricted) of prime numbers and divisors or the smallest divisor of the composite numbers has been prepared. Some of the most famous of these tables are:

Lehmer, D.N; Table of factors for the first ten millions numbers containing the smallest factor of every number not divisible by 2,3,5 and 7 between the limits "1" and "10,017,000" list of prime numbers from "1" to 10,006, 721[1].

Kulik, J. ph. Poletti.L and Porter. R. list of prime numbers of 11-th million (precisely: from 100006741 to 10999997)[2].

Preparing the prime numbers table and the table of factors have old antecedent. In chapter "5" of the book "Arithmetic" by Fibonacci (1202) a list of prime numbers from "11" to "97" has been given, and also he mentioned the factorization of numbers 12-100 to their factors. Cathaldy provided a table from the prime numbers smaller than 750 in his research about perfect numbers. Then the bigger tables had prepared gradually. Some characteristic of some of the most important tables are presented in the next page, except 3 tables that has been mentioned at the first[3].

Tables of Divisors

Divisors table

Date

Provider

1-2,856,000

1785

Felkel

1-3,000,000

1817

Burkhard

7,000,000-9,000,000

1863

Dazet

1-100,000,000

1867

Kulik

3,000,000-6,000,000

1883

Glishare

 

The propounded tables are based on the Burkhard, Kulik, Glishare and Dazet works. Preparing the big tables is very tedious and needs a big patient,
attempt, tolerance and every body who looks at these tables, couldnít restraint admiring the persons who has done this difficult job with no reward. Biller in his book "Entertainments in number theory" said "with seeing the attempt ion in this disordered world with war and continual conflict, it seems that the person who endowed the big part of their life to this work, must have found the secret of calmness and peace, that no wordily greedy personís can reach it absolutely". Antonio Felkel, whose name was mentioned in the previous pages, in 1776, computed divisors of all numbers up to 2,000,000. The first part of his table (up to 408000) was printed by the Vianís royal treasury but hadnít costume and in the Austria war with Turkish, was used for preparing cartridge. Felkel was planning to take back the rest of his table, but the treasury refused and since he disappointed to gain it, again he computed the divisors of the numbers from 408000 to 2856000. Philip Kulikís motivation in preparing the factorís tables is in conceivable. He had prepared this table lonely in 2o years. Lehmer verified the first volume of these 8 volumes in Royal academy in Wiena and described it in the preface of factorís table with enthusiasm and anxiety so that impresses the readers. The Kulikís table was used in preparing the table of "Kulik, Poletti and Forter" that we have mentioned before, and reformed its mistakes.

2.2. 2.2. Calculation of tables

According to our previous statements, it is clear that easily but with patience, itís possible to prepare the table numbers.

In fact, it took long many years to determine the prime numbers smaller than 10 million. These tables are printed and for each of the first 10 million numbers, the smallest odd divisor was written in front of them. Therefore, each number that nothing is written in front of it (has no divisor), it is prime number. Also some others have calculated the prime numbers between two big numbers (bigger than 10 million) for example, the prime numbers between 100,000,000 and 100,100,000. It is clear that these calculations are very detailed and long, because all numbers between these two numbers that are multiplicand of prime numbers smaller than approximate square root of 100 million should be omitted.

(It means the smaller prime numbers less than 10,000 that are nearly 2000 numbers).

Extension and completing the calculators help to speed of calculations. However it must be knotted to prevent the mistake of copying, and printing.

It is expected, this valuable work helps to improvement our knowledge about prim numbers greatly and this analysis (that we canít explain it, in this book).

However we must confess, there are a lot of ambiguous points despite so many attempts to recognition the prime numbers. But with patient researches that done about the prime numbers tables, it seems that some of these theorems will be confirmed by the mathematicianís reasoning. No doubt that this experimental and sensuous method, by some of the mathematician has been ursine successfully. And here, it isnít un-useful to present some of the researches to cleat the prime numbers recognition category.

2.3. 2.3. Stochasticís Theorem

One of the most attractive theorems that are resulted from these statistic researches is stochastic theorem. This theorem has perfect relation with sieve alternationís rule (that we have mentioned it before), But we can say that random theorem completes this rule in some cases.

It is clear that divisible numbers by prime divisors 2, 3 and 5 are repeated in every 30 successive numbers frequently. The numbers not divisible by 2, 3 and 5 are in one of below form:

30n+1              30n+7              30n+11

30n+13                        30n+17                        30n+19

30n+23                        30n+29                       

(Or: , ).

In the first ten numbers, There are two numbers and in the second ten numbers, four and in the third ten numbers, there are two numbers of these.

In this numbers, there are 26 numbers that are between "1" to "100" and 28 numbers between 101 to 200 and 26 numbers between 201 to 300 and these numbers are repeated successively in every next 100 numbers in this way.

Now, if we consider the prime numbers between 2 to 13, their multiplications are equal or 30030. Here again the same procedure is repeated but succession period is equal to 30030 instead of 30 and also if we devoid successive numbers to the packs containing 30 numbers, (the packs that have 8 un-divisible number by 2 and 3 and 5), the number of numbers that are divisible by 7, 11, 13 in these patches will be various. For the first ten packs (1 to 30 and 31 to 60 and Ö and 271 to 300) the numbering of these numbers are 3,1,1,2,3,2,3,2,3,1,... and in this way the number of numbers that are not divisible by 2,3,5,7,11,13 will be equal to 5,7,7,6,5,6,5,6,7 (that are calculated  from subtracting the previous values of 8). By the same way, various values will be calculated for the first 1001 packs of the 30 number namely up to 30030.

Only when we reach to this number, the previous values will be repeated prodigally. It means that there are 5 number between 30031 to 30060 that they are not divisible by 2,3,5,7,11,13 ( that equal the number of these numbers, when we consider the number between 1 to 30).

It seems that when want to increase the number of prime numbers, the length of succession period will grow rapidly. So if we want to search for the numbers that are not divisible by any of smaller prime numbers than 100, the succession period will be equal multiplication of these number that is a number with 30 digits approximately, and it is clear that the periodic method can not be used neither for the numbers that their primality can be investigated nor for the numbers bigger than one million that their smallest prime factor is near to 1000.

Now we can explain the random theorem: it concerns the distribution of the prime numbers in the consecutive packs of numbers. For example the distribution in the packs of 100- numbers, follows the random theorem, with considering the succession period for the smaller prime numbers. This will decrease the numbering of the prime numbers which exist probably.

If we limit the factors to 2, 3, 5 there will be 26 or 28 number in every 100 successive numbers, so that they arenít divisible by 2, 3, 5, and if we consider the factor 7 these number will even decrease. In other side it is possible to determine the average of the prime numbers which exist between every 100 numbers, for example the numbers between 1,000,000 and 1,100,000 and these numbers will be more than 7.

If the distributing of prime numbers were requited, there would be 7 prime numbers in every 100 number. But in fact despite the number 7 exist frequently, but the real number of prime numbers is often different from 7 in 100-number intervals and varies from 0 to 14.

Comparing the prime numbers with the obtained resorts, clarifies that in measure 100, in a box, there are 7 black balls in every 100 balls. It means 7 black balls and 93 white balls. Itís clear that difference of the number of prime numbers to their average is very less than the ratio that is shown here. Therefore, is necessary to pay attention to certain recurrence that is verified for small prime numbers.

The recurrence arranged this phenomenal. If we consider only prime factors 2, 3, 5, we will see that the number of numbers that are prime probably, (It means the numbers that are not divisible by 2, 3, 5) is equal to 26 or 28 in every recurrence 100 numbers. Therefore, is not necessary to consider a box with 100 balls. But we must consider a box for example which has 20 white balls against every 7 black balls (in measure 27 balls). It is possible to consider more prime numbers.

The multiplication of prime number up to 17 is equal to 510510 and up to 19 is equal to 9699690. The numbers which are under our consideration are between these two multiplications and if we consider prime number up to 17, another arrangement will be established. There are numbers between the first 510510 numbers that arenít divisible by prime number up to 17.

It means that there are 576016 numbers between 500000 numbers and it will be %18 of them. Now we consider the balls in every 18 balls, 7 black balls and 11 white balls and it is verified that the ratio calculated by theoretical hypothesis is very closer to the observed conditions.

Now we can arrange random theorem:

We consider the pack "T" which is randomly chosen from the successive 100 numbers for example between 2,000,000 and 2,100,000. The probability that the pack "T" includes "p" prime numbers is equal to the probability of existing "p" black balls in "q" balls in a box that has "n" black balls and
"" white balls. The numbers "q" and "n" are selected in this way: the number "q" is the possible number of number which can be in every 100 prime numbers, in clouding the numbers that are divisible by small prime number (up to 17), and the number "n" is the average number of prime numbers which are in every 100 number of studied pack.

It is clear that it is not correct to speak about the primality of a supposed number, because if a number be determined, it is easy to determine that it is prime or not. But we can speak about primality of a number if it is selected randomly with some condition. For example suppose that the number of a lottery winner is a prime number.

Here we use classification pattern to shorten numerical tables.

 

 The following table is designed for the interval (31,211]:
Note: The Composite numbers are bold.

+30

+60

+90

+120

+150

+180

 

7

11

13

7

37

67

97

127

157

187

 

7

49

 

 

11

41

71

101

131

161

191

 

11

77

121

 

13

43

73

103

133

163

193

 

13

91

143

169

17

47

77

107

137

167

197

 

17

119

187

 

19

49

79

109

139

169

199

 

19

133

209

 

23

53

83

113

143

173

203

 

23

161

 

 

29

59

89

119

149

179

209

 

29

203

 

 

31

61

91

121

151

181

211

 

 

 

 

 

 

? Multiplication table determining composite numbers in the interval (31,211]

2.3.1. Note. To eliminate composite number from the table it suffices to form the multiplication table of the interval under consideration.

Considering the multiplication table for the aforementioned interval 12 composite numbers are obtained which are to be eliminated from the table associated with the cycle 30s+p.

 

 

For the interval (211, 2311] the table of classes is as the following:

(210,p) =1

+210

+420

+630

+840

+1050

+1260

+1470

+1680

+1890

+2100

11

221

431

641

851

1061

1271

1481

1691

1901

2111

13

223

433

643

853

1063

1273

1483

1693

1903

2113

17

227

437

647

857

1067

1277

1487

1697

1907

2117

19

229

439

649

859

1069

1279

1489

1699

1909

2119

23

233

443

653

863

1073

1283

1493

1703

1913

2123

29

239

449

659

869

1079

1289

1499

1709

1919

2129

31

241

451

661

871

1081

1291

1501

1711

1921

2131

37

247

457

667

877

1087

1297

1507

1717

1927

2137

41

251

461

671

881

1091

1301

1511

1721

1931

2141

43

253

463

673

883

1093

1303

1513

1723

1933

2143

47

257

467

677

887

1097

1307

1517

1727

1937

2147

53

263

473

683

893

1103

1313

1523

1733

1943

2153

59

269

479

689

899

1109

1319

1529

1739

1949

2159

...

...

...

...

...

...

...

...

...

...

191

401

611

821

1031

1241

1451

1661

1871

2081

2291

193

403

613

823

1033

1243

1453

1663

1873

2083

2293

197

407

617

827

1037

1247

1457

1667

1877

2087

2297

199

409

619

829

1039

1249

1459

1669

1879

2089

2299

209

419

629

839

1049

1259

1469

1679

1889

2099

2309

211

421

631

841

1051

1261

1471

1681

1891

2101

2311

 

According to the tables, just mentioned, it follows that prime numbers are generated by the following general formula:

            (1)

In the above, is called the cycle constant and consider it to be equal to one of the numbers in the following sequence:

In (1) "p" is called the generating number. This definition justify adding a multiple of to "p" would result in new prime numbers. The positive integer depends on the cycle constant; i.e., if we consider the cycle constant as follows (is the prime number following)

The upper and lower bounds for are as follows:

We conclude that the set of prime numbers consists of a union of numerical cycles as described in (1). Each of these cycles is consisted of "n" numerical classes. The number "n" is in fact equal to the ratio of the distance of two consecutive cycles constant to the first one:

    (The number of classes)

According to this prime numbers could be regularized meaning that prime numbers associated with symbols "" or "" could be listed in a table with formulas of the form (1). For example, according to the former tables prime numbers of the cycle ,  for and could be written as follows :

2.3.2. Note. If in case the number of prime numbers of classes in a cycle is larger, it's better to use the symbol "", otherwise, the symbol "" is used.

As another example the cycle,, for are as follows:

 

Here, the following table is used to regularize prime number. This table could be extended to any given arbitrary cycle.

Prime numbers table in terms of consecutive cycles.


 

Table of prime numbers (H.M)

 

   

 

 

 

5

 

 

 

 

 

7

 

 

 

 

11

 

 

 

 

13

 

 

 

17

 

 

19

 

23

 

29

 

31

 

37

 

 

41

 

 

43

 

 

47

 

 

53

 

 

59

 

 

61

 

 

67

 

 

 

 

71

73

79

83

89

97

101

103

107

109

113

121

F

F

F

127

131

137

139

143

F

F

F

149

151

157

163

167

169

F

F

173

179

181

187

F

F

F

191

193

197

199

209

F

F

F

211

221

 

F

F

223

 

227

 

229

 

233

 

...

...

...

...

...

 

2.3.3.Note This table gives any prime less than, i.e. .

 

The following table (I) is designed for the number that are divisible by prime numbers of classes:

 

 

a: "2" , "3" and "5".                     b: from "7" up to "19".

c: from "23" up to "97".              d: greater than "100".

e: prime numbers.

Table (I)

N

A

b

c

d

e

Error controlling

2000000+

0001000

10001000

20001000

30001000

40001000

50001000

60001000

70001000

80001000

90001000

 

7334

7334

7332

7334

7334

7332

7334

7334

7332

7334

 

955

959

956

954

956

957

957

954

960

959

 

497

508

502

513

505

512

504

507

507

502

 

509

508

517

509

534

503

511

531

515

531

 

705

691

693

691

670

696

684

674

686

674

 

 

 

 

(+1)

(-1)

summand

73334

9567

5057

5168

6874

 

For every column of this table, a table with move detailed is prepared that we show that for column e, namely the column of prime numbers, (table (II) in next page). Every row of this tables justify with a pack of 10000 numbers like every row of the previous table.

Table (II)

 

1

2

3

4

5

6

7

8

9

10

11

12

13

0001000

10001000

20001000

30001000

40001000

50001000

60001000

70001000

80001000

90001000

 

 

 

 

2

1

 

 

2

2

1

 

 

2

2

2

3

3

5

2

2

3

4

5

2

3

7

8

9

4

8

5

6

7

9

5

13

10

9

14

13

10

12

13

9

15

25

20

13

17

26

23

25

14

17

17

10

22

23

25

14

23

18

23

25

22

13

17

15

22

11

13

12

16

18

20

16

15

11

4

11

12

11

14

11

10

10

2

8

6

11

7

7

3

3

6

2

1

5

3

 

1

3

3

3

 

2

 

 

1

2

2

 

1

1

 

 

1

summand

3

11

32

68

118

197

205

157

115

63

21

8

2

 

And this pack is divided to 100 packs of 100 numbers. The recorded numbers in various columns of the first row of this table, determine the number of prime numbers in packs of 100 numbers from these 100 packs (that started from 2000001). So there are three packs of these packs that every one includes 3 prime numbers. 7 packs include 4 prime numbers, 13 packs include 5 prime numbers. 25 packs include 6 prime numbers, 10 packs include 7 prime numbers, 13 packs include 8 prime numbers, 16 packs include 9 prime numbers, 10 packs include 10 prime numbers, 2 packs include 10 prime numbers, 2 packs include 11 prime numbers and at last one pack include 13 prime numbers and the next rows follow this order.

Investigating this table, show that some disorders exist in every row, about distribution of 100 numbers which include the number of prime numbers (from 1 to up 13), (average number that obtained from table (I) is 6.874).

First row has two maximum numbers (25 and 16).

Numbers of six-th column decrease unexpectedly for the rows 20001 and 70001 (13 and 14) and also in rows 00001 (10) and 40001 (14). But the seams show a kind of ordered distribution of prime numbers according to the first law of lap lass Ė gauss, with an important distribution when the number of prime numbers equals "1" in some packs of 100 numbers and in some other equals 13. But this distribution is result of the random that obtained for 1000 series, and probability of existing 17 prime numbers is in every series. Like a box which include 17000 balls that 6874 balls are black. The observed disorders are usual and it confirms the random theorem.

Now we search the obtained results from previous table, about 10000 packs of 100 numbers up to 10 million, namely numbers between 9000001 up to 10 million.

Table (III)

n

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

90 to 91

91 to 92

92 to 93

93 to 94

94 to 95

95 to 96

96 to 97

97 to 98

98 to 99

99 to100

6182

6245

6259

6223

6177

6271

6202

6208

6181

6134

 

 

 

 

 

1

 

1

4

2

2

3

6

1

3

3

1

4

21

20

26

21

24

13

25

17

15

21

54

60

48

53

56

54

55

60

67

65

124

126

98

103

105

124

111

122

113

104

176

151

161

161

189

157

176

173

180

188

186

195

228

215

200

194

212

197

208

209

179

189

190

195

166

206

170

167

178

178

131

127

127

114

120

139

112

133

119

115

82

78

77

73

84

59

83

68

70

67

32

29

24

24

41

32

34

40

28

33

9

15

12

12

6

10

14

17

15

10

2

6

6

3

3

8

4

2

3

5

 

2

1

2

 

2

1

 

2

 

 

 

1

 

 

 

 

1

1

summand

62082

2

29

203

572

1140

1712

2044

1818

1237

751

317

120

42

10

3

 

In the first of table (III) which is identified by , there are 100,000 units in every row, it means 1000 packs of 100 numbers, for example from  up to , then in column "n" the number of prime numbers of this interval is written, and in next columns that numbers from "1" up to "14" is written above them, the number of 100-number packs is identified which have "1" or "2",..., or "14" prime numbers. As it is found out from the sums, it seems that the distribution of prime numbers are arranged completely and almost, there is a result for 10 million like the same result for 3 million. Therefore, we can accept that a general rule exists.

It is surprising to use this statistics of the prime numbers about approximate distances, very big number like 100 million. The column is very interesting because it shows that how is the frequency of prime numbers near the indeterminate number "N". This column result that there is important diversity for big distances in relative to this theoretical frequency.

For distances of 100,000, the number of prime numbers that are in 10th million is equal to:

6182                        6245                        6223                        6177
6271                        6202                        6201                        6134

It seems that the first number is very small abnormally, but its second root hasnít big diversity in relative to average value so that it is possible to consider it as normal.

If we put 100,000 instead of 200,000, disordering will decrease so that if we add adjacent numbers two by two, will have:

12427, 12482, 12448, 12410, 12315

and with addition of three-to-three of these numbers (distances of 300,000):

18686                      18671                      18591                      (61314)

The obtained number will approach to each other continuously. Finally by adding           4-to-4 or 5-to-5:

24909                                      24858
31086                                      30996

2.4. 2.4. Another research on stochastic theorem

 Mr. Giuseppe Palama had presented and it led to searching about random theorem. The subject is definition of prime numbers in the form  and between 11000000 and 11100000.

Euler had shown that if integer number "n" chooses values, only numbers in  form are prime numbers. He had determined a lot of these values of "n" that the greatest of them is. This remark of Euler proves (Mr. Palamaís statistics), when he determined 23 prime numbers in  form between the numbers  and.

Also in every 100 packs of 1000 numbers there are in average 2.03 prime numbers with above form in distance 100000. We can determine easily number "N" that is indicator of the number of packs of 1000 numbers and include 0, 1, 2, 3, 4, 5, Ö prime numbers, so below table obtains:

Table (IV)

 

0

1

2

3

4

5

6

7

8

summand

7

25

38

23

3

4

0

0

0

100

Poisson

13.5

27.1

27.1

18

9

3.6

1.2

0.3

0.1

99.9

 

Also he mentioned a form of non Ė prime odd numbers in this distance but this form isnít complete.

In third row, we have recorded the theoretical values resulted form Poissonís relation.

It is clear that the concerned distribution proves the random rule. But it has more tendencies to a kind of arrangement in relation to random rule.

2.5. 2.5. Table of divisors of Burkhard

We express a skillful work from J.C.Burkhard in 1817 (divisors table), reprinted with permission from Gs. Carr from the famous book "Formulas and theorems in mathematics" (New York: Chelsea publishing company 1970) that it develops to "99000". This table calculates the least divisor of every integer number from "1" to "90000" that isnít multiple of 2 or 3 or 5 (except prime numbers "P" which put O (Zero) against them).

 A skillful work form J.C.Burkhard in 1817 that reprinted by printed by Permission from Gs. Carr from the famous book ďFormulas and Theorems in MathematicsĒ (New York: Chelsea publishing company 1970) you can see a part of this table in the next page.

 

Table (V)

 

00

03

06

09

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

69

72

75

78

81

84

87

01

07

11

13

17

 

19

23

29

31

37

 

41

43

47

49

53

 

59

61

67

71

73

 

77

79

83

89

91

97

0

0

0

0

0

 

0

0

0

0

0

 

0

0

0

7

0

 

0

0

0

0

0

 

7

0

0

0

7

0

7

0

0

0

0

 

11

17

7

0

0

 

11

7

0

0

0

 

0

19

0

7

0

 

13

0

0

0

17

0

0

0

13

0

0

 

0

7

17

0

7

 

0

0

0

11

0

 

0

0

23

11

0

 

0

7

0

13

0

17

17

0

0

11

7

 

0

13

0

7

0

 

0

23

0

13

0

 

7

31

0

0

7

 

0

11

0

23

0

0

0

17

7

0

0

 

23

0

0

0

0

 

17

11

29

0

7

 

0

13

7

31

19

 

0

0

0

0

0

0

19

11

0

17

37

 

7

0

11

0

29

 

23

0

7

0

0

 

0

7

0

0

11

 

19

0

0

7

37

0

0

13

0

7

23

 

17

0

31

0

11

 

7

19

0

43

17

 

11

0

0

0

0

 

0

0

7

0

31

7

11

7

0

0

29

 

13

11

0

0

0

 

0

0

19

7

0

 

17

0

11

13

41

 

7

0

37

11

7

13

7

29

0

19

0

 

41

0

7

11

0

 

0

7

0

31

11

 

0

23

0

7

0

 

0

37

13

19

47

11

37

0

0

0

11

 

0

7

0

0

7

 

0

13

41

0

0

 

31

11

0

17

47

 

0

7

11

0

0

0

0

31

0

23

7

 

0

0

13

7

0

 

0

17

11

0

43

 

7

0

0

37

7

 

17

0

0

0

11

19

0

0

7

0

31

 

0

0

0

0

47

 

13

0

0

17

7

 

0

0

7

0

0

 

11

31

17

0

0

43

13

0

23

0

0

 

7

0

19

0

0

 

11

0

7

41

13

 

0

7

19

0

0

 

0

13

29

7

0

0

47

0

0

7

0

 

0

0

0

0

31

 

7

0

0

11

59

 

37

17

0

11

29

 

41

23

7

0

13

7

0

7

0

11

0

 

0

41

0

0

19

 

0

0

31

7

0

 

0

0

17

0

0

 

7

11

0

0

7

0

7

0

13

0

0

 

0

0

7

23

13

 

19

7

0

0

29

 

47

0

0

7

17

 

23

19

0

13

0

0

0

11

17

0

0

 

61

7

11

0

7

 

47

29

37

13

23

 

43

0

31

0

11

 

0

7

19

0

67

59

0

0

19

0

7

 

0

47

23

7

11

 

53

37

0

19

0

 

7

13

0

0

7

 

31

0

71

0

29

0

11

0

7

0

0

 

0

11

61

0

0

 

0

0

13

0

7

 

53

43

7

0

13

 

0

0

0

11

17

23

0

13

0

29

0

 

7

59

17

11

0

 

0

0

7

0

11

 

13

7

73

29

23

 

53

0

0

7

0

11

17

0

6

7

11

 

13

19

0

37

0

 

7

0

0

23

0

 

73

11

0

13

0

 

59

0

7

0

0

7

0

7

0

59

0

 

71

0

0

13

0

 

17

0

11

7

0

 

0

0

0

23

0

 

7

0

13

0

7

0

7

0

11

17

13

 

0

37

7

19

0

 

29

7

17

61

0

 

0

0

59

7

0

 

11

0

41

0

0

37

67

0

0

31

0

 

11

7

13

29

7

 

11

53

0

0

17

 

0

0

0

0

19

 

0

7

0

29

0

0

19

0

0

0

7

 

0

31

0

7

0

 

13

0

0

11

0

 

7

53

13

11

7

 

19

29

0

37

23

0

13

0

7

11

0

 

73

0

0

17

0

 

0

19

0

0

7

 

0

0

7

67

0

 

0

11

0

0

0

71

29

37

73

13

0

 

7

0

0

41

17

 

0

11

7

47

0

 

29

7

0

17

0

 

0

0

0

7

13

53

0

11

0

7

0

 

23

0

11

47

79

 

7

17

0

29

31

 

41

0

0

0

11

 

13

0

7

19

0

7

31

7

13

47

19

 

0

0

0

0

11

 

23

0

0

7

79

 

11

0

0

43

37

 

7

61

17

13

7

29

7

0

31

0

23

 

0

11

7

0

0

 

0

7

0

13

0

 

19

0

11

7

31

 

67

0

0

11

59

19

 

 

01

04

07

10

13

16

19

22

25

28

31

34

37

40

43

46

49

52

55

58

61

64

67

70

73

76

79

82

85

88

01

03

07

09

13

 

19

21

27

31

33

 

37

39

43

49

51

57

 

61

63

67

69

73

79

 

81

87

91

93

97

99

0

0

0

0

0

 

7

11

0

0

7

 

0

0

11

0

0

0

 

7

0

0

13

0

0

 

0

11

0

0

0

0

0

13

11

0

7

 

0

0

7

0

0

 

19

0

0

0

11

0

 

0

0

0

7

11

0

 

13

0

0

17

7

0

0

19

7

0

23

 

0

7

0

17

0

 

11

0

0

7

0

0

 

0

7

13

0

0

19

 

11

0

7

13

0

17

7

17

19

0

0

 

0

0

13

0

0

 

17

0

7

0

0

7

 

0

0

11

0

29

13

 

23

0

0

0

0

7

0

0

0

7

13

 

0

0

0

11

31

 

7

13

17

19

7

23

 

0

29

0

37

0

7

 

0

19

13

7

11

0

0

7

0

0

0

 

0

0

0

7

23

 

0

11

31

17

13

0

 

11

0

0

0

7

23

 

41

7

19

0

0

0

0

11

0

23

0

 

19

17

41

0

0

 

13

7

29

0

0

19

 

37

13

7

11

0

0

 

7

0

11

0

0

0

31

0

0

47

0

 

7

0

17

23

7

 

0

0

0

13

0

37

 

7

31

0

0

0

43

 

0

0

29

0

0

11

41

0

23

13

7

 

11

0

7

0

17

 

43

0

0

0

0

0

 

13

11

17

7

31

0

 

29

13

0

0

7

23

0

0

7

63

29

 

0

7

11

19

0

 

0

17

0

7

0

0

 

0

7

47

19

13

0

 

43

0

7

11

0

13

7

29

13

0

11

 

0

0

53

31

13

 

0

43

7

47

23

7

 

29

0

0

0

19

11

 

0

0

0

31

23

7

19

41

0

7

0

 

13

11

23

47

0

 

7

19

11

0

7

0

 

0

0

0

0

23

7

 

59

11

0

7

13

0

0

7

11

0

47

 

0

61

0

7

0

 

37

0

19

23

11

13

 

0

53

0

0

7

0

 

19

7

17

0

0

29

0

0

0

19

0

 

0

0

0

29

27

 

11

7

13

0

0

0

 

31

17

7

13

0

0

 

7

61

0

0

17

0

11

13

59

31

19

 

7

29

0

61

7

 

0

0

43

0

19

0

 

7

0

11

17

0

29

 

13

41

0

23

0

53

43

0

17

11

7

 

31

0

7

11

41

 

0

0

0

0

0

0

 

59

0

13

7

0

0

 

31

43

0

13

7

37

13

0

7

0

17

 

0

7

13

0

0

 

0

11

0

7

0

0

 

11

7

0

0

0

13

 

17

0

7

0

19

0

7

11

41

0

13

 

17

23

0

0

0

 

0

13

7

29

59

7

 

0

19

23

11

0

0

 

0

17

11

67

0

7

0

0

0

7

37

 

0

0

0

0

11

 

7

29

23

31

7

0

 

67

0

19

0

0

7

 

0

37

0

7

29

11

0

7

0

37

0

 

11

0

0

7

19

 

13

0

0

0

0

0

 

0

11

0

0

7

0

 

0

7

43

71

0

17

0

17

31

41

0

 

29

0

11

0

0

 

17

7

0

11

0

47

 

61

0

7

31

0

37

 

7

23

41

11

0

0

37

19

43

13

11

 

7

0

0

59

7

 

41

47

17

0

0

11

 

7

23

29

0

0

11

 

0

13

0

43

73

67

0

0

19

0

7

 

0

11

7

53

0

 

0

23

11

17

43

29

 

0

0

67

7

13

0

 

0

11

0

0

7

13

0

47

7

43

0

 

0

7

0

79

13

 

31

0

0

7

11

0

 

23

7

37

0

11

0

 

73

19

7

41

47

31

7

67

0

0

71

 

13

0

17

0

0

 

11

41

7

0

0

7

 

17

27

62

0

73

47

 

11

83

19

0

13

7

11

0

0

7

23

 

19

0

29

13

17

 

7

0

0

0

7

13

 

47

79

11

0

0

7

 

0

0

0

7

43

0

0

7

0

11

41

 

0

89

0

7

0

 

0

17

13

0

0

72

 

19

0

31

12

7

79

 

23

7

61

0

11

19

59

13

29

0

43

 

0

0

19

0

0

 

0

7

0

73

37

22

 

11

0

7

0

0

17

 

7

0

0

0

0

43

0

11

47

67

0

 

7

0

0

19

7

 

0

0

0

83

17

43

 

7

0

13

11

0

23

 

0

31

11

13

0

0

13

0

0

23

7

 

0

0

7

0

11

 

0

0

37

0

63

17

 

0

0

0

7

19

13

 

83

0

17

0

7

11

 

  

 

 

02

05

08

11

14

17

20

23

26

29

32

35

38

41

44

47

50

53

56

59

62

65

68

71

74

77

80

83

86

89

03

09

11

17

21

 

23

27

29

33

39

 

41

47

51

53

57

 

59

63

69

71

77

 

81

83

87

89

93

99

7

11

0

7

13

 

0

0

0

0

0

 

0

13

0

11

0

 

7

0

0

0

0

 

0

0

7

17

0

13

0

0

7

11

0

 

0

17

23

13

7

 

0

0

19

7

0

 

13

0

0

0

0

 

7

11

0

19

0

0

11

0

0

19

0

 

0

0

0

7

0

 

29

7

23

0

0

 

0

0

11

13

0

 

0

0

0

7

19

29

0

0

11

0

19

 

0

7

0

11

17

 

7

31

0

0

13

 

19

0

7

0

11

 

0

7

0

29

0

11

23

0

17

13

7

 

0

0

0

0

0

 

11

0

0

0

31

 

0

7

13

0

7

 

0

0

0

0

0

0

13

0

29

17

0

 

0

11

7

0

37

 

0

0

17

0

7

 

0

41

29

7

0

 

13

0

0

0

11

7

0

7

0

0

43

 

7

0

0

19

0

 

13

23

7

0

11

 

29

0

0

19

31

 

0

0

0

0

7

0

7

0

0

7

11

 

23

13

17

0

0

 

0

0

0

13

0

 

7

17

23

0

0

 

0

0

7

0

0

0

19

0

7

0

0

 

43

37

11

0

7

 

19

0

11

7

0

 

0

0

17

0

0

 

7

0

0

0

0

0

0

0

41

0

23

 

37

0

29

7

0

 

17

7

13

0

0

 

11

0

0

0

13

 

11

19

29

7

41

0

0

0

13

0

0

 

11

7

0

53

41

 

7

17

0

0

0

 

0

13

7

0

29

 

17

7

19

11

37

0

31

11

0

0

7

 

13

0

0

0

0

 

0

0

53

11

0

 

0

7

43

0

7

 

0

0

17

37

0

59

0

13

37

11

0

 

0

43

7

0

11

 

23

0

0

0

7

 

17

0

53

7

0

 

0

11

13

0

17

7

11

7

0

23

13

 

7

0

0

0

0

 

41

11

7

0

0

 

0

23

11

43

0

 

37

47

53

59

7

3

7

0

11

7

0

 

0

19

43

11

23

 

0

0

0

61

0

 

7

0

41

17

11

 

0

0

7

67

0

11

0

17

7

53

0

 

0

29

0

0

7

 

11

47

0

7

67

 

0

11

19

13

17

 

7

0

0

0

0

0

0

0

0

29

0

 

0

11

47

7

0

 

71

7

0

31

13

 

0

61

37

11

0

 

0

13

0

7

11

0

0

0

47

13

17

 

0

7

73

0

19

 

7

0

0

53

11

 

23

31

7

41

19

 

0

7

0

17

0

0

13

17

31

41

7

 

0

17

13

43

0

 

0

0

0

0

0

 

0

7

0

63

7

 

13

0

11

0

0

41

0

19

23

61

31

 

0

0

7

17

0

 

13

19

11

0

7

 

59

67

47

7

43

 

0

31

0

53

13

7

0

7

0

0

0

 

7

13

0

23

17

 

79

0

7

13

0

 

11

0

0

0

0

 

11

61

0

19

7

0

7

23

17

7

0

 

11

61

0

47

13

 

31

0

0

0

79

 

7

0

0

0

0

 

0

29

7

11

19

0

0

11

7

17

19

 

0

0

0

0

7

 

0

41

13

7

0

 

19

0

0

0

13

 

7

0

71

83

61

0

0

0

13

11

0

 

17

0

0

7

11

 

37

7

0

23

17

 

0

13

67

71

0

 

43

11

0

7

0

23

11

31

0

0

41

 

13

7

17

0

43

 

7

11

0

29

0

 

0

17

7

31

0

 

0

7

0

0

59

0

0

13

11

0

7

 

0

0

59

11

71

 

0

61

23

0

0

 

0

7

17

19

7

 

31

43

13

0

0

11

53

0

0

0

13

 

71

23

7

29

0

 

11

13

83

0

7

 

0

11

0

7

41

 

0

59

0

0

0

7

19

7

0

0

53

 

7

11

0

13

31

 

19

17

7

0

61

 

13

0

0

11

0

 

17

83

0

0

7

37

7

0

79

7

37

 

0

0

0

89

53

 

0

0

41

17

11

 

7

0

0

13

0

 

0

19

7

0

0

0

29

59

7

37

11

 

0

79

0

0

7

 

0

23

0

7

13

 

17

0

0

0

47

 

7

13

11

89

17

0

 

This table calculates the least divisor of every integer number from 1 to 90000 that isnít multiple of 2, 3 or 5. The first two digits of integer number (from left to right) are in row numbers from bold kind and the last two digits are in left column. For example:


[1]. Carnegie Institution of Washington Publication 105, 1909 (165, 1914)

[2]. From tables of manuscript, Amsterdam 1951

[3]. Historical subject of this part is based on the first volume of "History of theory of numbers" Dickson, "primary numbers" by Biller. Table in the next page is from the last book.