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)*.

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)*.

* 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.

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.

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.

Fri Feb 20 21:45:30 EST 1998