By Paul Hoffman
An advent to the delights and demanding situations of recent arithmetic.
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Extra resources for Archimedes' Revenge: The Joys and Perils of Mathematics
The ciphers used today in the military and in private industry barely resemble the ciphers of yesteryear and are generally much harder to break. And yet, despite these advances, in many situations the newfangled mathematical codes are not useful, and some age-old ciphers remain immune to state-of-the-art codecracking techniques. , when Julius Caesar reportedly used a naively simple substitution cipher, in which each letter was replaced by the letter that followed it alphabetically by three places.
If contemplating unproved conjectures is not your thing, perhaps thinking about specific numbers is. But don't make Hardy's mistake of prematurely dismissing a taxicab number as uninteresting. I was recently on a long plane flight, absorbed in a novel, when the fidgeting fellow in the next seat made an awkward stab at initiating conversation: "We are on flight 407. " "Nonsense," I replied, without looking up from my reading. "The number isn't dull at all. It's quite interesting. " The man looked Prime Prostitution 47 me as if I were crazy, but he took out a pad and started scribbling numbers.
X P. From the form of the number Q. it is obvious that no integer from 2 to P divides evenly into it; each division would leave a remainder of 1. If Q is not prime, it is evenly divisible by some prime larger than P. On the other hand, if Q is prime, Q itself is a prime larger than P. Either possibility implies the existence of a prime larger than the largest prime. This means, of course, that the concept of "the largest prime" is a fiction. But if there's no such beast, the number of primes must be infinite.
Archimedes' Revenge: The Joys and Perils of Mathematics by Paul Hoffman
Categories: Mathematical Physics