Herman Baer on Ruth-Aaron pairs

Moledet, July 14, 2005

Dear Mr. Peterson,

In Paul Hoffman's book on Erdős "The man who loved numbers" I found a remark on "714 and 715" and began to ponder on solutions.

Later on, my eldest grandson, several months ago, provided me from the internet with your record on this problem of June 1997, of the end of which you state: "Comments are welcome".

Making use of this offer I send you the enclosed list of 26 + 10 + 2 = 38 solutions.

It developed, more or less, like this:

Up to 10000 I checked a tabulation of factorization of numbers; further on, missing such, I employed in essence the sieve of Eratosthenes, writing down a section of successive numbers, noting in turn all prime dividers 2,3,5,7; 11,13,17,19,23,29 ... ... and so on, stopping short if the square-root of the limiting number; eventually using pocket calculator HP, as well as for completing the factorization to build the sum of all its prime factors.

Beside I found a method to combine two pairs of solutions to a bigger one, in a limited number of cases:

Let a<b, then

(a+1)b - (b+1)a = b - a = D;

If D/(a+1)b and D/(b+1)a, then

c = (b+1)a/D, c+1 = (a+1)b/D

S(c) = S(a) + S(b) - S(D) is a new solution.

In this way it is possible to combine

Nrs(1,2) → Nr3, [Nrs(1,3) → Nr2]

Nrs(8,9) → Nr21, Nrs(9,10) → Nr19

Nrs(20,21) → Nr36, Nr(13,14) → Nr100~

Nr(29,30) → Nr135~

The first 4 I encountered only when the results already were known to me; on the contrast the 3 last ones without knowing the results in advance.

I extended my Marathonic search up to the prime number 42437, about the average distance above the solution Nr36.

In addition to the neighboring pairs I also observed coincidences of more distant pairs of equal sums of prime factors, as well as neighboringor distant sequences of such sums.

Statistically their density increases with increasing numbers, because the sum of their prime factors grows only slowly in comparison.

This only apparently contradicts the fact, that the density of neighboring pairs in general decreases, though with sporadic higher densities.

So far, sincerely yours Hermann Baer


(Complete) List of solutions  a < 42000 (Three more distant)

Nr

S(a)

P(a)

a

Diff to Next = Δ

a+1

P(a+1)

S(a+1) = S(a)

1

5

5

5

3

6

2*3

5

2

6

2^3

8

7

9

3*3

6

3

8

3*5

15

62

16

24

8

4

18

7*11

77

48

78

2*3*13

18

5

15

52

125

589

126

2*32*7

15

6

29

2*3*7*17

714

234

715

5*11*13

29

7

86

22*3*79

948

382

949

13*73

86

8

33

2*5*7*19

1330

190

1381

113

33

9

32

24*5*19

1520

342

1521

32*132

32

10

35

2*72*19

1862

629

1863

34*23

35

11

100

47*53

2491

759

2492

22*7*89

100

12

44

24*7*29

3248

937

3249

32*192

44

13

45

33*5*31

4185

6 (twins #1)

4186

2*7*13*23

45

14

141

3*11*127

4191

1214

4192

25*131

141

15

75

5*23*47

2405

155

5406

2*3*17*53

75

16

150

23*5*139

5560

399

5561

67*83

150

17

160

59*101

5959

908

5960

23*5*149

160

18

122

32*7*109

6867

1413

6868

22*17*101

122

19

40

23*32*5*23

8280

183

8281

72*132

40

20

54

3*7*13*31

8463

2184

8464

24*232

54

21

39

32*7*132

10647

1764

10648

23*113

39

22

205

3*23*179

12351

2236

12352

26*193

205

23

532

29*503

14587

2345

14588

22*7*521

532 (Max)

24

107

22*3*17*83

16932

148

16933

7*41*59

107

25

79

23*5*7*61

17080

1410

17081

19*29*31

79

26

93

2*5*432

18490

1960

18491

11*412

93

27

421

2*52*109

20450

4445

20451

3*17*401

421

28

401

5*13*383

24895

1747

24896

26*389

401

29

193

2*7*11*193

26642

7  (twins #2)

26643

3*83*107

193

30

66

34*7*47

26649

1799

26650

2*52*13*41

66

31

144

25*7*127

28448

361

28449

32*29*109

144

32

117

33*11*97

28809

4210

28810

2*5*43*67

117

33

149

7*53*89

33019

4809 (Max)

33020

22*13*127

149

34

211

22*72*193

37828

53 (twins #3)

37829

11*19*181

211

35

93

33*23*61

37881

3380

37882

2*13*31*47

93

36

64

113*31

41261

41262

2*3*13*232

64

 

 

100~

181

24*32*5*31*131

29239290

29239291

7*11*13*23*127

181

135~

252

22*52*11*13*41*193

101429900

101429901

35*47*83*107

252

 

40~

78

2*3*11*312

63426

63427

7*13*17*41

78

 

This last solution I happened to find only today by "outside" generalized combination of 

Nr13, d1 = 1

And      31365 = 32*5*17*41 → 69

            31372 = 22*11*23*31 → 69

            d2 = 7

D = a*d2 - b* d1 = 2070 = 2*32*5*23

c = a(b + d2)/D, c+1 = b(a + d1)/D


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