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sci.math FAQ: Unsolved Problems


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Archive-name: sci-math-faq/unsolved
Last-modified: February 20, 1998
Version: 7.5

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                              Unsolved Problems


                                       
Does there exist a number that is perfect and odd?

   A given number is perfect if it is equal to the sum of all its proper
   divisors. This question was first posed by Euclid in ancient Greece.
   This question is still open. Euler proved that if N is an odd perfect
   number, then in the prime power decomposition of N, exactly one
   exponent is congruent to 1 mod 4 and all the other exponents are even.
   Furthermore, the prime occurring to an odd power must itself be
   congruent to 1 mod 4. A sketch of the proof appears in Exercise 87,
   page 203 of Underwood Dudley's Elementary Number Theory. It has been
   shown that there are no odd perfect numbers < 10^(300).
   
Collatz Problem

   Take any natural number m > 0.
   n : = m;
   repeat
   if (n is odd) then n : = 3*n + 1; else n : = n/2;
   until (n==1)
   
   Conjecture 1. For all positive integers m, the program above
   terminates.
   
   The conjecture has been verified for all numbers up to 5.6 * 10^(13).
   
      References
      
   Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem
   E16.
   
   Elementary Number Theory. Underwood Dudley. 2nd ed.
   
   G.T. Leavens and M. Vermeulen 3x+1 search programs ] Comput. Math.
   Appl.
   
   vol. 24 n. 11 (1992), 79-99.
   
Goldbach's conjecture

   This conjecture claims that every even integer bigger equal to 4 is
   expressible as the sum of two prime numbers. It has been tested for
   all values up to 4.10^(10) by Sinisalo.
   
Twin primes conjecture

   There exist an infinite number of positive integers p with p and p+2
   both prime. See the largest known twin prime section. There are some
   results on the estimated density of twin primes.
-- 
Alex Lopez-Ortiz                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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