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)

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

Nr13, d_{1 = 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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