Ivars Peterson's MathTrek

June 30, 1997

# Playing with Ruth-Aaron Pairs

On April 8, 1974, Henry (Hank) Aaron hit his 715th major league home run, surpassing the previous mark of 714 career home runs long held by baseball great Babe Ruth. Understandably, the event received considerable coverage in newspapers and magazines and on television.

However, those reports invariably overlooked the mathematical aspects of that achievement, particularly the curious properties of the two numbers 714 and 715. It took the efforts of mathematicians Carol Nelson, David E. Penney, and Carl Pomerance at the University of Georgia to call attention to this facet.

Notice that 714 = 2 x 3 x 7 x 17 and 715 = 5 x 11 x 13; so 714 x 715 = 2 x 3 x 5 x 7 x 11 x 13 x 17. In other words, the product of the two consecutive whole numbers 714 and 715 is equal to the product of the first seven prime numbers!

Pomerance and his colleagues wondered whether there were other pairs of consecutive numbers whose product is also the product of the first k primes. The first few instances are easy to find: 1 and 2 (1 x 2 = 2), 2 and 3 (2 x 3 = 2 x 3), 5 and 6 (5 x 6 = 2 x 3 x 5), 14 and 15 (14 x 15 = 2 x 3 x 5 x 7), and 714 and 715. The mathematicians then used a computer to search for such pairs, going as far as products of the first 3,049 primes (numbers up to 6,021 digits long). They found no more examples in that range.

Footnote: On April 26, 1974, Aaron hit his 15th grand slam home run, breaking the old National League record of 14.

And there's more. Notice that the sum of the prime factors of 714 is 2 + 3 + 7 + 17 = 29, and the sum of the prime factors of 715 is 5 + 11 + 13 = 29. How often do two consecutive numbers have prime factors that add up to the same total?

Pomerance and his coworkers conducted another computer search, looking for such pairs up to a value of 20,000.

Here are the first few examples:

Numbers Sums
5, 6 5 = 2 + 3
8, 9 2 + 2 + 2 = 3 + 3
15, 16 3 + 5 = 2 + 2 + 2 + 2
77, 78 7 + 11 = 2 + 3 + 13
125, 126 5 + 5 + 5 = 2 + 3 + 3 + 7
714, 715 29
948, 949 86
Pomerance called these pairs Aaron numbers, and he speculated that such pairs become less frequent as their size increases. However, he didn't have a mathematical proof quantifying their scarcity.

With a week of the appearance of these results in the Journal of Recreational Mathematics, Pomerance received a telephone call from the legendary mathematician Paul Erdos (see Paul Erdos: An Infinity of Problems), who offered to show him how to prove the conjecture. Pomerance invited Erdos to Georgia, and the meeting resulted in a joint paper. It was the first of more than 40 papers that the two mathematicians would write together.

During his lifetime, Erdos collaborated with so many mathematicians that these efforts have been captured in something called the Erdos number (see Groups, Graphs, and Paul Erdos). Erdos is assigned the number 0. People who have coauthored a paper with him are given the number 1. People who have coauthored a paper not with Erdos but with someone who coauthored a paper with Erdos get the number 2, and so on.

Pomerance notes that many years after his initial collaboration with Erdos, the University of Georgia awarded honorary degrees to both Erdos and Aaron. On that occasion, he asked the recipients to autograph a baseball for him. In effect, Hank Aaron joined the ranks of those having Erdos number 1!

Footnote: 714 + 715 = 1429. Notice anything about 1429? It's a backwards-forwards-sideways prime, which means that 1429, 9241, 1249, 9421, 4129, 4219 are all prime numbers. What about 1492? That was the year that Columbus stumbled upon America.