rcs@xmission.com wrote:
The only theorem I know of is that a rational number always has a finite representation in factorial base; therefore e is irrational.
How does that follow? The existence of an infinite representation for e in factorial base doesn't prove there isn't also a finite representation.
Back in the day, I was looking through the file of paper tapes in the PDP1 room (this served as the software library), and I came across one labeled simply "e". The handwriting might have been either Eric Jensen or Bill Ackerman. It was only a couple of folds, a few hundred characters.
A few years ago I wrote the following obfuscated c program, which prints a thousand digits of e: void main(){int j,k,l[1002];for(j=1776;j;j--){for(k=0;k<1001;k++){l[k+1]+= 10*(l[k]%j);l[k]/=j;}l[0]++;}for(j=0;j<1001;j++){printf("%d",l[j]);if(!j) printf("%c",46);}printf("\n");} I'm especially proud of operating on an uninitialized array, just to drive readers insane. "It *can't* work, but it does." Can anyone write an equally short c program to print the first thousand decimal digits of any other interesting irrational number, such as pi, phi, zeta(3), cos(x)=x, ln(2), gamma, or sqrt(2)? (An irrational number contrived to have its first thousand digits have some simple pattern (e.g. all 0) is not interesting.) (Okay, nobody knows if gamma is irrational, but it's close enough for me.)