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Math Trek

Playing with Ruth-Aaron Pairs

By
4:00pm, August 2, 2005

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 t

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