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Math Trek

Playing with Ruth-Aaron Pairs

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4:00pm, August 2, 2005
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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 the 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 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 integers 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 additional examples in that range. It’s now conjectured that 714 and 715 is the last pair of consecutive integers whose product is the product of the first k primes for some k.

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, 65 = 2 + 3
8, 92 + 2 + 2 = 3 + 3 = 6
15, 163 + 5 = 2 + 2 + 2 + 2 = 8
77, 787 + 11 = 2 + 3 + 13 = 18
125, 1265 + 5 + 5 = 2 + 3 + 3 + 7 = 15
714, 7152 + 3 + 7 + 17 = 5 + 11 + 13 = 29
948, 94986

Such numbers are now known as Ruth-Aaron pairs. Pomerance speculated that these pairs become less frequent as their size increases. However, he didn’t have a mathematical proof quantifying their scarcity.

Within a week of the appearance of these results in the Journal of Recreational Mathematics, Pomerance received a letter from legendary mathematician Paul Erdös, who offered to show him how to prove the conjecture. Pomerance invited Erdös to Georgia, and the meeting resulted in a joint paper giving the proof, which was published in 1978. It was the first of more than 40 papers that the two mathematicians would write together.

During his lifetime, Erdös collaborated with so many mathematicians that these efforts have been captured in something called the Erdös number (see Groups, Graphs, and Erdös Numbers at Groups, Graphs, and Erdös Numbers). Erdös 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 Erdös but with someone who coauthored a paper with Erdös get the number 2, and so on.

Pomerance notes that many years after his initial collaboration with Erdös, the University of Georgia awarded honorary degrees to both Erdös and Aaron. On that occasion, he asked both recipients to autograph a baseball for him. In effect, Hank Aaron joined the elite ranks of those having Erdös number 1! Even though Aaron doesn't have a joint paper with Erdös, he does have a joint baseball.

In the meantime, the search for Ruth-Aaron pairs has continued. In 1996, John L. Drost reported that a computer search yields 149 pairs below 1,000,000. The list can be found at http://www.trottermath.net/numthry/ravdata.html. Somewhat later, Joe K. Crump developed a method for generating Ruth-Aaron pairs. It hasn't yet been proved, however, that the number of such pairs is infinite.

Drost went on to look for other analogously interesting sets of numbers. For example, he found a triple with equal prime sums: 417162 = 2 x 3 x 251 x 277; 417163 = 17 x 53 x 463; and 417164 = 2 x 2 x 11 x 19 x 499. All have prime sums of 533. Is there another such triple? No one knows.

Going back to 714 and 715, there’s still more. Note that 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.

Originally posted: 6/28/97

Updated: 8/6/05

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