Carnal Reason

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

Edna St. Vincent Millay

Euclid is remembered as the father of axiomatic geometry, and hence of modern mathematics. He also proved that there are infinitely many primes. His proof is the model of elegance and economy of means. Suppose there are only finitely many known primes, and let P be their product. P+1 is larger than and hence distinct from any of the known primes. P+1 is not divisible by any known prime, since dividing P by any of the known primes leaves a remainder of 1. Either P is itself prime, or it has a prime factor larger than any of the known primes used to construct P.

It is interesting to consider other approaches to proving that there are infinitely many primes. Planet Math presents Fürstenberg’s proof, which uses a clever topological argument.

The great Euler devised a profound and beautiful approach, described by Gregory Chaitin in one of his very interesting books. I reproduce Chaitin’s exposition of Euler’s proof on a separate page, here.

Posted on May 30th, 2005 by pwyll
Filed under: science