two hundred thirty-nine

The MIT Artificial Intelligence Laboratory Memo 239 (entitled HAKMEM) of February 1972 listed various properties of the integer 239, some more interesting than others, such as: that 239 needs the maximum number of powers in Waring’s problem for squares, cubes and fourth powers; that it appears in Machin’s formulaMathworldPlanetmathPlanetmath for π:


that since 2392=2×134-1, the fraction 239132 is a convergentMathworldPlanetmathPlanetmath for the continued fractionMathworldPlanetmath of 2; that since π(1500)=239 (with π(x) being the prime counting function; etc. At the time, the memo couldn’t have mentioned that 239 pounds is Homer Simpson’s weight (as established by at least two episodes of The Simpsons).

The prime 239, like many in its vicinity, is a Chen primeMathworldPlanetmath. As a Sophie Germain primeMathworldPlanetmath, it begins a Cunningham chainMathworldPlanetmath of just length 2 (which ends with 479). On the plane of Eisenstein integersMathworldPlanetmath, 239 is an Eisenstein primeMathworldPlanetmath (its real partMathworldPlanetmath being of the form 3n-1 and it not having an imaginary part), and it is also a Gaussian primeMathworldPlanetmath (its real part being of the form 4n-1); these two properties it has in common with all real primes p11mod12. Much more rare is that it is the third Newman-Shanks-Williams prime (the ninth is more than 1098).

Title two hundred thirty-nine
Canonical name TwoHundredThirtynine
Date of creation 2013-03-22 16:57:56
Last modified on 2013-03-22 16:57:56
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Feature
Classification msc 11A99
Synonym two hundred and thirty-nine