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Numbers Too Large (and Small) to Count
At one point in the children’s novel “The Phantom Tollbooth,” its protagonist, Milo, sets out to reach infinity. When he abandons the quest as hopeless, he is advised “Infinity is a poor place.”
“Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity,” by Antonio Padilla, holds a different view. As the author shows, infinity can be terrifying, but it is filled with an endless amount of numbers.
This is not a book just about numbers. It is about the relationship between numbers and physics, and how the world works at both the largest and the smallest scales. The numbers Padilla examines are fantastic in two senses. They are so extreme as to challenge belief and they are so extravagant as to seem fancy. And they define how the universe works.
Most people have trouble with numbers. They easily visualize up to twelve. Once beyond 100 numbers kind of blur together. There is a difference between a one in 500 chance of something happening and a one in a million chance, but most people do not really understand it. Or the difference between a million and a billion.
So for years, like all of you, I’ve been idly wondering if it’s true that For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d1+ε, where d denotes the product of the distinct prime factors of abc. I figure that’s a question that occurs to everyone in the small hours of the morning, now and again. Once, when I was stuck in the métro during a wildcat strike, I reckoned I’d figured out a marvellous little proof. But I scribbled it down on the back of a pack of Gitanes, shoved it into the Pile of Papers, and never found it again.
I want to take note of John Nash’s 
