The real numbers we are so accustomed to (ℝ) extends from -∞ to +∞. However, it is overlooked that real numbers subtly have 2 components: a natural (ℕ) number component {0, …, ∞}; and a direction (the positivity or negativity of the absolute number). Numbers vs Quantities There is a subtle but profound difference between…
Polignac’s Conjecture, and the Twin Prime Conjecture are widely considered to remain unproven since first proposed in 1849. We provide a proof of the conjectures by defining a prime in terms of its lack of divisibility and express the probability of its occurrence based on its indivisibility. Inadvertently, a more generalised form of Polignac’s Conjecture…
Given a prime P(m): Example: For P(m) = 11,
Goldbach’s conjecture states that: Every even integer greater than can be expressed as the sum of two prime numbers. It’s apparent that the recurring theme of is expressed thrice in this statement for good reason: they are related. Even refers to divisibility by . So, suppose we could extend this to any arbitrary divisibility and…
Consider that primes are the dissonant instances within harmonic sets of those dissonant instances. That is, the nth prime begins a harmonic set across . Suppose: the positive integer set, is the prime number set, is the nth prime, and where is a prime number, and is the set of all products of in .…
A type of thing is a consistent invocation of sensations that a person identifies. A type of thing will be referred to formally as a quality set, in that it is a set of distinctive qualities one identifies as being fulfilled. For example, a cat is quality set regarding the slenderness of the nose, the…