Problem of the Week 874

Factoring a Factorial

Write 56! as a product of 56 positive integers, so that the smallest is as large as possible. No proof is required -- do the best you can.

For more of a challenge, do the same for 300! as a product of 300 integers.

Source: Factoring Factorial n, by Guy and Selfridge, after a problem by Erdos, Selfridge, and Guy (Amer Math Monthly, p. 766-767, Oct 1998).

Integer of the Week: 134670080641

n = 134670080641 = 211873 x 635617

Notice that:

n is a 2-strong pseudoprime
n is a 3-strong pseudoprime
n is a 5-strong pseudoprime
n is NOT a 6-strong pseudoprime

In short, the first "strong pseudoprime witness" to the compositeness of n is 6, a nonprime. I found this by looking through a database computed by Daniel Bleichenbacher of pseudoprimes up to 10^17.

Local News

The Macalester Programming Team of Tamas Nemeth, Karim Farouki, and Vahe Poladian finished second in the Assoc. of Computing Machinery regional programming contest, and will be traveling to the Netherlands in the spring for the international finals. I believe this is the first year that the finals will be held in a country other than the U.S. Results for our region:

                                    Number    Penalty Points 
      Team                          Solved    (based on time)
------------------------            ------    --------------- 
 1. U. Nebraska at Lincoln A           5       656         
 2. Macalester                         5       665         
 3. U. Nebraska at Lincoln B           5       737         
 4. South Dakota Sch. Mines & Tech     5       786        
 5. Michigan Tech U.                   5       822  
10. Macalester B                       4

A mathematical joke

This is from Al Taylor, whose book on Fair Division Algorithms was just reviewed in the Amer. Math. Monthly. Al calls this a real-world fair division joke. It's from the venerable TV program, The Honeymooners.

  Ralph:   "Norton, I'll tell you what Alice did. She put two potatoes
            on the table -- a big one and a small one -- and then she 
            took the big one!"

  Norton:  "What would you have done?"

  Ralph:   "Well -- I'd have taken the small one."

  Norton:  "You got the small one."

© Copyright 1998 Stan Wagon. Reproduced with permission.