Roderick MacPhee

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

Collatz Conjecture

A sequence of form $2^{m}x+b$ 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

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 $m\in \mathbb {N}$ Has a prime between itself and $m+4\lfloor {\sqrt {m}}\rfloor +4$

Goldbach's conjecture

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

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.

Twin prime conjecture

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.