dansmath > lessons > number theory > primes
 
 
dan's prime page (updated 11/02)
also see supercomposites - the opposite of prime!
 
 

First, the basic definition:

How do you tell "primes" from "non-primes" ?

Table of the First 400 Prime Numbers : (17 being, of course, the most random)

Simple Mathematica Command: Table[Prime[k], {k, 1, 400}]

Do the primes ever stop? They seem to be getting farther apart...

Table of World Record Primes (by year, Electronic Computer Age)

2^32,582,657 - 1 is currently the largest known prime (discovered September 2006)

This has 9,808,358 digits; we are very close to the ten million digit barrier!

It's a prime example of a Mersenne prime, one of the form 2^p - 1 , where p is a prime number.

If n is odd & composite then so is 2^n - 1: n = ab --> 2^n - 1 = (2^a - 1)[2^(n-a) + 2^(n-2a) + . . . + 2^a + 1]
Ex: 2^9 - 1 = (2^3 - 1)(2^6 + 2^3 + 1), so 2^9 - 1 = 511 = (8-1)(64+8+1) = 7 * 73 is composite.
But 11 is prime, and 2^11 - 1 is not; although you can't factor it with this formula.
 
The notation Mp means 2^p - 1. (If p is not prime then 2^p - 1 isn't either.) M2 = 3, M3 = 7, M7 = 127 ;
The largest known prime from 1876-1950 was M127 = 170,141,183,460,469,231,731,687,303,715,884,105,727.
 
Here is a table of the "World Record Primes" by year from Chris Caldwell's Prime Pages,
www.utm.edu/research/primes/ . . . Click here for early history of prime size

 Number  Digits  Year  Machine  Prover
180(M127)^2+1  79  1951  EDSAC1  Miller & Wheeler
 M521  157  1952  SWAC  Robinson (Jan 30)
 M607  183  1952  SWAC  Robinson (Jan 30)
 M1279  386  1952  SWAC  Robinson (June 25)
 M2203  664  1952  SWAC  Robinson (Oct 7)
 M2281  687  1952  SWAC  Robinson (Oct 9)
 M3217  969  1957  BESK  Riesel
 M4423  1,332  1961  IBM7090  Hurwitz
 M9689  2,917  1963  ILLIAC 2  Gillies

 M9941  2,993  1963  ILLIAC 2  Gillies
 M11213  3,376  1963  ILLIAC 2  Gillies
 M19937  6,002  1971  IBM360/91  Tuckerman
 M21701  6,533  1978  Cyber 174  Noll & Nickel (H.S.students)
 M23209  6,987  1979  Cyber 174  Noll
 M44497  13,395  1979  Cray 1  Nelson & Slowinski
 M86243  25,962  1982  Cray 1  Slowinski
 M132049  39,751  1983  Cray X-MP  Slowinski
 M216091  65,050  1985  Cray X-MP  Slowinski
 391581*
2^216193 - 1
 65,087  1989  Amdahl 1200  Amdahl Six

M756839  227,832  1992  Cray-2 Slowinski & Gage
M859433  258,716  1994  Cray C90 Slowinski & Gage
M1257787  378,632  1996  Cray T94 Slowinski & Gage
M1398269  420,921  1996 Pentium 90 Armengaud, Woltman
M2976221 895932 1997 Pentium 100 Spence, Woltman
M3021377 909,526 1998 Pentium 200 Clarkson, Woltman, Kurowski
M6972593 2,098,960 1999 Pentium 350 Hajratwala, Woltman, Kurowski
M13466917 4,053,496 2001 AMD 800 Cameron, Woltman, Kurowski
M20996011  6,320,430  2003  Pentium 2G  Michael Shafer & GIMPS
M24036583  7,235,733  2004  P4 2.4GHz  Josh Findley & GIMPS
M25964951  7,816,230  2005  P4 2,4GHz  Nowak using GIMPS
M30402457  9,152,052  2005 P4 3.0GHz  Cooper, Boone, GIMPS
M32582657  9,808,358 2006   P4 3.0GHz  Cooper, Boone, GIMPS et al.

All of the Mersenne records were found using the Lucas-Lehmer test and the other two were found using Proth's Theorem (or similar results).
The Amdahl Six is J. Brown, C Noll, B Parady, G Smith, J Smith and S Zarantonello.
 
 
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