  We prove the twin prime conjecture and the generalized conjectures of
Kronecker and Polignac.
  Key to the proofs is a new theoretical sieve that combines two concepts that
go back to Eratosthenes: the 'sieve' filtering a finite set of numbers and the
'hydra' as a representation of infinity. Using functional programming notation
(and a reference implementation in R) we define a data structure 'hydra' that
partitions the infinite set of numbers into a finite set of partitions. On top
of this data structure we define an algorithm 'split' which sub-partitions each
partition using the modulus prime function. Hydra splitting along the natural
sequence of primes is a recursive version of wheel factorization. Hydra
recursion allows elementary proofs of some statements about primes. We consider
one new and two classical proof structures. Using these proof methods we find
that hydra recursion proves the infinity of twin primes. Then we show how to
use specific selections of primes to create hydras that contain pairs of
partitions with arbitrary even distance, which proves Maillet's conjecture and
the stricter 'consecutive existence conjecture'. Together with our proof
methodology we obtain elementary proofs of Kronecker's and Polignac's
conjecture.
