MERCY COLLEGE - THE DEAN'S COLUMN
Appreciating Geometric Relationships: The Ruth-Aaron Numbers
By Alfred Posamentier, Ph.D.

In my most recent Book, Mathematical Curiosities: A Treasure Trove of Unexpected Entertainments (Prometheus Books, 2014), I felt it was time to apply mathematics to our national pastime, baseball. However, not in the way that we would expect by considering the geometric applications folder on the baseball field and the action that occurs on the field but from two numbers there for a long time help define the primary goal of the game. For many years the goal for most home-run-hitting baseball players was to reach, or surpass, the longtime record for career home runs set by Babe Ruth at 714 home runs. On April 8, 1974, the Atlanta Braves’ slugger Hank Aaron hit his 715th career home run. This prompted Carl Pomerance, a mathematician at the University of Georgia, to popularize the notion – through the suggestion of one of his students. The amazing relationships about these two numbers, 714 and 715, that are related by dint of the fact that they are two consecutive numbers whose prime factor sums are equal leads to many other arithmetic discoveries which can be used to enrich a secondary school classrooms instruction and enchant students with the endless curiosities to be discovered within the realm of mathematics. Let’s consider these two numbers now.

714 = 2 ´ 3 ´ 7 ´ 17, and 2+3+7+17 = 29,

715 = 5 ´ 11 ´ 13, and 5+11+13 = 29.

We can extend the list of Ruth-Aaron numbers, if we consider only numbers with distinct prime factors. In this case we have the following pairs:

(5, 6), (24, 25), (49, 50), (77, 78), (104, 105), (153, 154), (369, 370), (492, 493), (714, 715),

(1682, 1683), (2107, 2108).

If multiplicities are not counted (so that a factor of 23 counts only a single 2), we get the following pair:

(24, 25): 24 = 2 ´ 2 ´ 2 ´ 3, and 2+3 = 5,

25 = 5 ´ 5, and 5 = 5.

If we consider repeating prime factors, then the following numbers would also qualify as Ruth-Aaron numbers:

(5, 6), (8, 9), (15, 16), (77, 78), (125, 126), (714, 715), (948, 949), (1330, 1331).

(24, 25): 8 = 2 ´ 2 ´ 2, and 2+2+2 = 6,

9 = 3 ´ 3, and 3+3 = 6.

There are also pairs of unusual Ruth-Aaron numbers, where the sums of all the factors are equal, and where the sums of the factors, without repetition of each of the factors, are equal. One such care is: (7129199, 7129200), where 7,129,199 = 7·112·19·443, and 7,129,200 = 24·3·52·13·457.

Without repeating the factors of each of the numbers:

7+11+19+443 = 2+3+5+13+457 = 480,

The sums taking into account the repetition of the factors:

7+11+11+19+443 = 2+2+2+2+3+5+5+13+457 = 491.

We can extend the Ruth-Aaron pairs to consider Ruth-Aaron triplets, that is, three consecutive numbers, where the sums of the factors are equal. Here is one such triplet: (89,460,294, 89,460,295, 89,460,296)

89,460,294 = 2·3·7·11·23·8,419

89,460,295 = 5·4,201·4,259

89,460,296 = 2·2·2·31·43·8,389

The sums of the factors are as follows:

2 + 3 + 7 + 11 + 23 + 8,419 = 5 + 4,201 + 4,259 = 2 + 31 + 43 + 8,389 = 8,465.

This is the first example without repetition!

Another such Ruth-Aaron triplet is the following: (151,165,960,539, 151,165,960,540, 151,165,960,541)

151,165,960,539 = 3·11·11·83·2,081·2,411

151,165,960,540 = 2·2·5·7·293·1,193·3,089

151,165,960,541 = 23·29·157·359·4,021

The sums of the factors are:

3 + 11 + 83 + 2,081 + 2,411 = 2 + 5 + 7 + 293 + 1,193 + 3,089 = 23 + 29 + 157 + 359 + 4,021 = 4,589.

Here we have, yet, another Ruth-Aaron triplet, but with repeating the factors: (417,162, 417,163, 417,164)

417,162 = 2·3·251·277

417,163 = 17·53·463

417,164 = 2·2·11·19·499

The sums of the prime factors are:

2 + 3 + 251 + 277 = 17 + 53 + 463 = 2 + 2 + 11 + 19 + 499 = 533

The last Ruth-Aaron triplet so far to have been found is: (6,913,943,284, 6,913,943,285 6,913,943,286)

6,913,943,284 = 2·2·37·89·101·5,197

6,913,943,285 = 5·283·1,259·3,881

6,913,943,286 = 2·3·167·2,549·2,707

The sums of the prime factors are:

2 + 2 + 37 + 89 + 101 + 5,197 = 5 + 283 + 1,259 + 3,881 = 2 + 3 + 167 + 2,549 + 2,707 = 5,428.

Who would have thought that the two home-run kings would make a contribution to mathematics? By the way, we should remind ourselves that this discussion began with 714 and 715, and we take the sum of these numbers 714 + 715 = 1,429, which is a prime number, as is its reversal 9,241. Furthermore, other arrangements of this number are also prime numbers: 4,219, 4,129, 9,412 1,249.

Interestingly, Pomerance found that up to the number 20,000 there are 26 pairs of Ruth-Aaron numbers, the largest of which is (18,490; 18,491). One of the most famous mathematicians of the 20th Century, Paul Erdös (1913-1996), proved that there are an infinite number of Ruth-Aaron numbers. You might now want to challenge her students to search for others!#

*Alfred S. Posamentier is Dean of the School of Education, and Professor of Mathematics Education, Mercy College, New York. He was Distinguished Lecturer at New York City College of Technology of CUNY and is Professor Emeritus of Mathematics Education at The City College of CUNY.*