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;
if (n is odd) then n : = 3*n + 1; else n : = n/2;
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.