---------- Forwarded message ---------- From: <rcs@xmission.com> Date: Wed, Feb 11, 2015 at 2:57 PM Subject: Re: [math-fun] Factorial base To: math-fun@mailman.xmission.com Cc: rcs@xmission.com

The only theorem I know of is that a rational number always has a finite representation in factorial base; therefore e is irrational.

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. Curious, I printed it out, and then ran the assembler on it. Indeed, it promptly printed out a few thousand digits of e. I examined the code more closely, and discovered it created a factorial base representation of (the fractional part of) e, one digit per machine word, and then did a simple decimal conversion, multiplying the factorial base representation by decimal 10, and printing the integer "carry" that falls off the front end; rinse, repeat. The program, in assembly language, easily fit on one page. A gem.

Rich

The program was most likely written by the late Richard P. Howell, one of the noteworthy PDP-1 folks in the late 60's. I seem to recall he had some extremely clever program for calculating e. I don't remember ever examining it. Bill