Today's numbers are 714 and 715
Wow, two numbers in one post! Taken individually, neither 714 nor 715 is especially remarkable. Together, however, they are connected by one of the most delightful coincidences in recreational mathematics. On April 8, 1974, baseball player Hank Aaron hit his 715th major league home run. The prior record of 714 was held by Babe Ruth. Understandably, the event received considerable coverage in newspapers and on TV.
However, mathematicians Carol Nelson, David E. Penney, and Carl Pomerance, who all worked at the University of Georgia, found another fascination for these numbers.
Namely, they noted that 714 x 715 is the product of the first seven prime numbers!
714 = 2 x 3 x 7 x 17
715 = 5 x 11 x 13
Thus, 714 x 715 = 510,510 = 2 x 3 x 5 x 7 x 11 x 13 x 17
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 enough to find:
1 x 2 = 2
2 x 3 = 2 x 3
5 x 6 = 2 x 3 x 5
14 x 15 = 2 x 3 x 5 x 7
714 x 715 = 2 x 3 x 5 x 7 x 9 x 11 x 13 x 17
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 didn't find any additional examples.
It is now conjectured that 714 and 715 are the last pair of consecutive integers whose product is the product of the first k primes for some k. To this day, the conjecture remained an open problem.
Another interesting fact they found is that the sum of the prime factors of 714 is 29, and 715 is the same:
2 + 3 + 7 + 17 = 29
5 + 11 + 13 = 29
Again, Pomerance and his colleagues wondered if there were other numbers that had this property and again a computer was used to search for such pairs up to 20,000. Here are the first few examples. Note that repeated prime factors are counted with multiplicity:
5 -> 5 = 5 and 6 -> 2 + 3 = 5
8 -> 2 + 2 + 2 = 6 and 9 -> 3 + 3 = 6
15 -> 3 + 5 = 8 and 16 -> 2 + 2 + 2 + 2 = 8
77 -> 7 + 11 and 78 -> 2 + 3 + 13 = 18
125 -> 5 + 5 + 5 = 15 and 126 -> 2 + 3 + 3 + 7 = 15
714 -> 2 + 3 + 7 + 17 = 29 and 715 -> 5 + 11 + 13 = 29
948 -> 86 and 949 -> 86
Such numbers are now known as Ruth-Aaron pairs after the two baseball players.
Pomerance and his colleagues published their results in the Journal of Recreational Mathematics where they conjectured that Ruth-Aaron pairs become increasingly sparse as numbers grow larger.
After publication, Pomerance received a letter from the legendary mathematician Paul Erdös, who offered to show him how to prove the conjecture. Erdös was invited 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 the two would write together.
During his lifetime, Erdös collaborated with so many mathematicians that these efforts have been captured in something called the Erdös number. Erdös is assigned the number 0. People who have coauthored a paper with him have an Erdös number of 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, Emory University awarded honorary degrees to both Paul Erdös and Hank Aaron. On that occasion, he asked both recipients to autograph a baseball for him, so it can be said that Hank Aaron has an Erdös number of 1.
