HS #51 2019.11.14
The Beauty of Mathematics
This past summer while enjoying Thursday Street Performers on 8th Street, I stopped into Readers World and found myself captivated by Jim Holt’s, “When Einstein Walked with Godel.” I have seldom read a book that so skillfully interweaves history, biography, and mathematics.
One theme running through it is the beauty of mathematics. The great mathematician, G.H. Hardy, declared that beauty, not usefulness, is the true justification for mathematics. Bertrand Russell observed, "Mathematics possesses not only truth but supreme beauty."
How can mathematics be beautiful? Anyone who asks that question has never seen the Mandelbrot Set – certainly the most intricate and stunning thing in mathematics. Yet even simple things can contain hidden beauty. Case in point: consider the prime numbers, those like 2,3,5,7,11,13,17,19,23 that have no factors. (15, for example, is not prime since 15 can be divided by its factors 3 and 5.)
These innocuous primes hide deep mysteries. For example, every even number ever checked is the sum of two primes (12=5+7, 20=17+3). Although no exception has ever been found, is this true for all even numbers? No one has been able to prove it.
But the most elegant result involving the primes dates back to antiquity. Although they thin out as they grow in size, Euclid proved that there is no largest prime – that is, there are an infinite number of primes. His proof is the mathematical equivalent of Hamlet’s Soliloquy or Pachelbel’s Canon or Michelangelo’s “David.” And amazingly, it can be understood by anyone willing to put in a bit of effort. Are you game? Let’s start.
First a bit of motivation. This past summer a tech friend explained to me that the security of email messages, credit cards, and bank statements relies on big prime numbers. Each user (person or computer) has their own personal prime number as part of their security key. That requires a lot of primes! Are there enough to go around? And the prime number needs to be large enough that others can’t easily discover it. So it’s reassuring to know that there is an inexhaustible number of big primes.
Second, a bit of detail work: Consider (2x3x5)+1=31. Notice that 31 does not have the first three primes 2 or 3 or 5 as factors, because dividing 31 by any of them will result in a remainder of 1. Check it out. Similarly, 2, 3, 5, and 7 are not factors of (2x3x5x7)+1=211; for example, 211 divided by 7 gives 2x3x5=30 with a remainder of 1.
Another detail: Notice that if a counting number (1,2,3,4,5, . . ) is not prime, then it is the product of primes. For example, 15=3x5, 84=2x2x3x7. This fact is called the Fundamental Theorem of Arithmetic because it essentially shows that the primes are the building blocks of all the counting numbers. (What are the prime factors of 70?)
OK, let’s get started on the clever proof that Euclid discovered 2300 years ago. We begin by assuming for sake of argument that there are just a finite number of primes. Then there is a last and largest one – we’ll call it N. So let’s form a new number, much bigger than N, by multiplying all the primes together and then adding 1. We’ll call it P. So P=(2x3x5x7x11x13x17x . . . xN)+1.
Now we assumed that N was the largest prime number, but look at the fix that P puts us in: Either P is a prime number or P is not a prime number. (That’s from Aristotle’s logic.) If P is a prime number, then since P is larger than N, N is NOT the largest prime number after all. So our initial assumption was wrong.
If P is not a prime number, then, according to the Fundamental Theorem of Arithmetic, P can be written as the product of prime factors. But notice, just as 31 divided by 2, 3, or 5 leaves a remainder of 1, so also P divided by ANY of the entire list of primes all the way up to N leaves a remainder of 1. So if P is not prime, then the prime factors of P must be larger than N. Again, our initial assumption was wrong.
Since assuming that there is a largest prime number, N, leads inexorably to the contradictory conclusion that N is not the largest prime, that initial assumption must have been wrong. Thus, there is no largest prime – they go on forever.
How cool is that! Give yourself a pat on the back.
