User:Roderick MacPhee

A sequence of form $2^mx+b$ follows the same iteration pathway through the Collatz function that $b$ has until $2^m$ gets used up as the lead coefficient.

Legendre's conjecture, states there is always a prime between $n^2$ and $(n+1)^2$ . This is the same as saying that for any $m\backslash in\backslash mathbb\{N\}$ Has a prime between itself and $m+4\backslash lfloor\backslash sqrt\; m\backslash rfloor+4$

Goldbach's conjecture is p+q=2n has solution where p,q are primes. You can also state it, as all composites are the arithmetic mean, of 4 not necessarily distinct semiprimes. Either (p,2n-p) is a pair that works or distances n,-n congruent modulo p can be ruled out.

Grimm's conjecture is: a set of composite numbers, has a bijective mapping for prime divisors. This only works if $g\_n$ the nth prime gap, is less than n.

The twin prime conjecture is that there infinitely many twin prime pairs (pairs of primes that differ by 2). Because:

$(6a+1)(6b+1)=6(6ab+a+b)+1$and$(6a-1)(6b-1)=6(6ab-a-b)+1$

as well as $(6a-1)(6b+1)=6(6ab-b+a)-1$ we can say natural numbers not of these forms must exist or twin prime conjecture would be false.