Discrete Number System Hypothesis

The Query

The Fundamental Theorem of Arithmetic states that all integers (above 1) can be factored into prime numbers. However, we build number systems through a Linear Combination of numbers. For example, the popular decimal system uses a Linear Combination of 1, 10, 100, 1000, et cetera; more succinctly, 1 and 10 (will get back to why). Another example is the binary system, ever important to computers, that uses a Linear Combination of 1 and 2. An effective property of these number systems is that their structure is very regular. What if we were to construct a number system that did not have such regularity?

Instead, let's imagine a number system that comprises a Linear Combination of a few randomly selected natural numbers. What would be the lowest natural number such that all further numbers are contiguous? More formally, this is written as

L ⊂ ℕ, S = { n∈ℕ | n = ∑_b∈La_b*b }, ∃x∈ℕ, ({ n∈ℕ | n≥x } ⊆ S)⇒min(x)=?

Examples

• prime: L = { n∈ℕ | n ∈ prime } ⇒ S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … } ⇒ min(x) = 1
• decimal: L = { 1, 10 } ⇒ S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … } ⇒ min(x) = 1
• binary: L = { 1, 2 } ⇒ S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … } ⇒ min(x) = 1
• L = { 3, 5, 7 } ⇒ S = { 3, 5, 6, 7, 8, 9, 10, 11, 12, … } ⇒ min(x) = 5
• L = { 3, 5 } ⇒ S = { 3, 5, 6, 8, 9, 10, 11, 12, … } ⇒ min(x) = 8
• L = { 2, 9 } ⇒ S = { 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, … } ⇒ min(x) = 8
• L = { 4, 9 } ⇒ S = { 4, 8, 9, 12, 13, 16, 17, 18, 20, 22, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, … } ⇒ min(x) = 42

Identity Property

Since 1 multiplied by any number is that number again, called identity, then 1∈L⇒min(x)=1 and is thus trivial. We will ignore these going forward.

The Power of Two

Since 2 multiplied gives rise to all even numbers, 2∈L⇒even⊆S and transitions the discussion to the odd numbers present in L. More specifically we can derive min(x) entirely based upon the lowest odd number.

∃y∈odd, y∈L∧2∈L ⇒ min(x)=min(y)-1

This is also trivial, and we will ignore these going forward.

On Equivalence

Since some numbers are a multiple of other numbers, having both in L will yield the same S. For example, (L={ 2 }∨L={ 2, 4 })⇒even⊆S and thus yield the same min(x). All choices of L that yield the same S are considered equivalent and will be treated as just the choice of L that has the least elements comprising it.

Application

Quantum mechanics has determined that electrons sit at unique and consistent distances away from the nucleus (called Energy Levels). These Energy Levels contribute to the precise distances between atoms while they are bonded together into molecules. They also play a key factor in the precise distances between molecules when they form materials, which can then be visible to the naked eye in large enough quantity.

Would we, then, be able to assign natural numbers to these distances and build a number system exclusively with those numbers? Would all real distances in existence above the min(x) of such a number system be present in that number system, such that they are then considered discrete distances? How would that affect contemporary continuous distance theories?