A loser who apparently knows more about Grimm's conjecture, twin prime conjecture, Goldbach's conjecture, Collatz conjecture, Legendre's conjecture, and more.

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

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

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 ${\displaystyle g_{n))$ 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:

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