Do any of you like baseball? Although I am not a lover of sports, I still want to tell you a mathematical story, starting from a home run.
In 1974, Henry Aaron got his 715th home run. This broke the record 40 years ago, which was created by Babe Ruth. From that day, everyone in America were talking about it for a few months. Yes, it is ‘714 vs 715’.
Everyone were crazy about it, even mathematicians. A professor called Carl Pomerance thought about what is special about this two numbers. Because of his habit, he factorise their product and found something interesting: 714*715=2*3*5*7*11*13*17, which is equal to the product of the first seven primes. He thought that, if P(k) is the product of the first k primes, is it common that it equals to the product of two consecutive numbers. He used a computer to calculate this, and only found 5 examples where k<3050: 1*2, 2*3, 5*6, 14*15, and 714*715. That means if there exists another example, the two numbers should be greater than 10^6021. Therefore, he created a conjecture that 714*715 is the last example. Until now, it is not proved. You may try if you have time.
By the way, after he found this, he asked another professor about this. That professor asked the question in class and a student called Jeremy Jordan found another surprise: The two numbers share the same prime factors sum. Now, it is called Ruth-Aaron pair.
After that, Carl Pomerance, Carol Nelson and David Penney wrote an essay about it and it was published in the Journal of Recreational Mathematics. They suggested that Ruth-Aaron pairs have density 0 in all natural numbers. A few days later, they received a letter from Paul Erdos. He said that it could be proved and wanted to meet them. After their meeting, it changed Pomerance’s mathematical life. He published 20 essays with Paul Erdos later on.
In 1995, Paul Erdos and Henry Aaron got honour degrees in the same university. Pomerance met them and introduced one another. He also asked them to sign their names on the same baseball. That makes Henry Aaron’s Erdos number 1. Pomerance stores this baseball in a glass box until today. However, the signs are already invisible due to fading of ink.
