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### Subject: sci.math FAQ: Unsolved Problems

This article was archived around: 17 Feb 2000 22:55:51 GMT

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

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