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

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

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