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Lehmer Factoring Machines
Photoelectric,Chain, Gear
Manufacturer D.H. Lehmer Identification,ID X85.82 Date of first manufacture 1932 Number produced 1 Estimated price or cost  location in museum  donor
 Photoelectric machine  D.H. Lehmer
 Chain machine  " ... chain machine  replicated at the (Boston) Computer Museum by Roberto Canepa, Andrew Kristoffy and Richard Rubinstein."
The above two machines were combined on this page
as the principles of operation are identical, even though the mechanical details differ. Contents of this page:
 Photo
 Placard
 Architecture
 Special Features
 Historical Notes
 This Artifact
 Interesting Web Sites
 Other information
Photoelectric Factoring Machine
Bicycle Chain Factoring Machine  reproduction

The Factoring Machines did NOT use the technique of Eratosthenes The Lehmers were after much bigger fish  please see
 THE MECHANICAL COMBINATION OF LINEAR FORMS a theoretical discussion of the Chain Prime Number Sieve Machine by D. H. LEHMER
 HUNTING BIG GAME IN THE THEORY OF NUMBERS a popular discussion on humting primes by D. N. LEHMER
 Lehmer Sieves by Dr. Mike Williams
 D. H. Lehmer's Number Seives pages 3 & 4, from The Computer Museum Reports Volume 4

from http://www.netlib.org/nadigesthtml/91/v91n21.html From: Robert D. Silverman
Jim Roth
Date: Thu, 23 May 1991 01:49:49 GMT
Subject: D. H. Lehmer
It is with a very great deal of sadness that I must report that D. H. Lehmer passed away last night.
I am deeply saddened by this loss as Professor Lehmer's work has been a source of personal inspiration for me. I consider him to be the father of computational number theory; my major field of interest.
Bob Silverman
Mitre Corporation
Professor Lehmer worked on (among other things) the solution of sets of linear Diophantine equations via Chinese Remainder Theorem. This involved building a specialpurpose computer which had many "delay lines" (recirculating loops) of primeinteger length. When the numbers in all of the loops lined up (as in a Las Vegas slot machine hitting a jackpot) a bell would ring and the data would be dumped and printed out, then reloaded into the loops. The first of these machines was purely mechanical; later on (1965?) he bought surplus radar delay lines and worked out a purely electronic numbertheory machine along the above lines, which he talked about at UCLA. He claimed that his technique got answers hundreds of times faster than an IBM 704 would have. Does anyone know whether he ever got a special VLSI chip built for such purposes?
David L. Elliott
Washington University
There is a photograph and explanation of Dr. Lehmer's factoring machine in the Dover book "Recreations in the Theory of Numbers" by Albert Beiler. This machine had many gear driven wheels which would be halted when sets of holes in them lined up, triggered by a photocell. It was called a photoelectric number sieve.
Apparently it was an amazing, though tempermental, device, and was inadvertantly "jammed" by a local Ham radio operator whose transmissions triggered the photocell at times.
The book also mentions Lehmer's first factoring machine  a bicycle sprocket and chain contraption!
Will future generations look back similarly at our current massively parallel computers doing quantum chromodynamics calculations?
Jim Roth
Digital Equipment Corporation
from Mike Albaugh, Nov 22, 2017, talking about an end of life IBM computer at Berkeley, CA ... I was told that while the 7094 (7090?) had been purchased, all the peripherals (and presumably the 7044) had been rented. That was the explanation I was given for the "brain in a bucket" still being in the computer center, where its one user was D.H. Lehmer.Operators described to me how he and his wife would come in and "play a console duet" on the address and data switches, entering the code for some longrunning mathematical exploration. He would come back days (weeks?) later and read the results from core.
When he relinquished the machine, the Cal Computer Club was asked if we were interested, and an IBM CE was asked for his opinion of its condition. His report was that "someone" (Lehmer?) had apparently not been interested in floating point or IO, so had scrounged SMS cards to keep the integer stuff running. It was declared hopeless.
Mike

 YouTube LaFarr Stuart talking about the Lehmer Sieves 2002
 LaFarr's explaination of Factoring by recognizing perfect squares which was the actual usage of the devices in the Museum.
 Norman Hardy has computational formulas, C Code, and worked examples
 LEHMER CONFERENCE
 The Prime Pages
 D.H. Lehmer's Number Sieves  article in "The Computer Reports"  10 megabyte .pdf
 “Unprogramming” the ENIAC: Lehmer Child’s Play  2017
 A weekend off. The first extensive numbertheroetical computation on the ENIAC  2016, via Paul McJones.
"Recreations in the Theory of Numbers"  See page 239 for Lehmer gear machine
https://www.amazon.com/RecreationsTheoryNumbersDoverRecreational/dp/0486210960
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Updated May 15, 2017