Ruth-Aaron pairs

On May 25, 1935, at Forbes Field in Pittsburgh, Babe Ruth hit his 714th (and last) home run, setting a record for major league baseball which would stand unchallenged for nearly 40 years. The record was broken on April 8, 1974 by Hank Aaron who hit his 715th home run. (He went on to hit a total of 755 home runs. This record has since been broken by Barry Bonds with 762 home runs, and those three: Ruth, Aaron, Bonds, are the only three players ever to hit more than 700 home runs in major league baseball.)

Not long after Aaron’s 715th homer, Carl Pomerance, then at the University of Georgia, and two others, noticed two facts about the numbers 714 and 715:

  1. The two numbers between them factored into the first seven primes:

    714 = 2cdot 3cdot 7cdot 17

    and

    715 = 5cdot 11cdot 13.

  2. The sums of the prime factors of each were the same:

    2+3+7+17 = 5+11+13 = 29.

Pomerance and his colleagues wrote a paper called “714 and 715” which he later described as “humorous” discussing these numbers. This paper, had the effect of catching the eye of Paul Erdős, who suggested that he visit Pomerance in Georgia, thus starting a long and fruitful collaboration between the two (at least 40 joint papers).

In honour of the two baseball players, two consecutive integers the sum of whose prime factors (with multiplicity) are equal, are called Ruth-Aaron pairs. It’s not hard to set up a brute force program to list them:

sage: def S(n): return sum(i*j for i,j in factor(n))
....:
sage: for k in range(1,100000):
....:     if S(k)==S(k+1):
....:         print k
....:
5
8
15
77
125
714
948
1330
1520
1862
2491
3248
4185
4191
5405
5560
5959
6867
8280
8463
10647
12351

… and so on. This sequence is A039752 in the OEIS.

It is not yet known whether there are infinite Ruth-Aaron pairs, but it is known, thanks to Pomerance and Erdős, that they are quite rare; in fact Pomerance proved that for any integer x, the number of integers n<x for which S(n)=S(n+1) was bounded as

displaystyle{Oleft(frac{x(loglog x)^4}{(log x)^2}right)}

and that the sum of the reciprocals was bounded.

In 1995 Emory University awarded honorary degrees to both Aaron and Erdős. Pomerance, who was then at Emory, asked for a baseball – several had been provided for Aaron to sign – and asked both Aaron and Erdős to sign it. “I joke that Aaron should have Erdős-number 1 since, though he does not have a joint paper with Erdős, he does have a joint baseball.”

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