We introduce a novel geometric framework for number theory by mapping natural numbers to infinite-dimensional vectors based on their prime factorizations. This representation, denoted as \( \mathbb{N}^\infty \) , allows us to define an inner product, norm, distance, and angle in a structured way. We explore how these geometric constructs capture fundamental number-theoretic properties, enabling a new perspective on prime structure relationships.
Number theory is traditionally studied through algebraic and combinatorial techniques. However, in this paper we suggest that geometric interpretations of number-theoretic structures may provide new insights.
Therefore, we propose a vector space representation of natural numbers, where each number is associated with a vector of prime exponents. This approach allows us to introduce inner products, distances, and angles, thereby transforming multiplicative number theory into a geometric framework.
The structure we introduce, \( \mathbb{N}^\infty \), provides:
This geometric embedding suggests new ways to study primes, factorization behavior, and number-theoretic functions using vector space techniques.
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers, up to the order of the factors. Mathematically, this is expressed as:
\( n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \)
where \( p_1, p_2, \ldots, p_k \) are distinct consecutive prime numbers and \( a_1, a_2, \ldots, a_k \) are non-negative integers.
We define the space \( \mathbb{N}^\infty \) as an infinite-dimensional discrete space, where each natural number is mapped to a vector of non-negative integers representing its prime exponents.
Inspired by Dirac's braket notation in quantum physics, we define a ket representation function \( | \rangle : \mathbb{N} \rightarrow \mathbb{N}^\infty \) as:
\( | n \rangle = [a_1, a_2, a_3, \ldots] \)
while the bra \( \langle | : \mathbb{N} \rightarrow \mathbb{N}^\infty \) is defined as:
\( \langle n | = \\ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \end{bmatrix} \)
where \( a_i \) is the exponent of the \( i \)-th prime number in the factorization of \( n \). This results is a countably infinite-dimensional vector with only finitely many nonzero components.
Although braket notation was originally developed in the context of Hilbert spaces in quantum physics, we adopt it here for its clarity and expressiveness in representing natural numbers as exponent vectors. Since our vectors lie in \( \mathbb{N}^\infty \) with only finitely many non-zero components, the inner product and related operations remain well-defined without requiring the completeness of a Hilbert space.
Examples:
We introduce a shorthand for the above by removing the trailing zero's to get:
Try it yourself:
We define an inner product between two natural numbers \( n \) and \( m \) using their exponent coordinate vectors \( | n \rangle = [a_1, a_2, \ldots] \) and \( | m \rangle = [b_1, b_2, \ldots] \):
\( \langle n | m \rangle = \sum_{i=1}^\infty a_i b_i \)
This inner product is well-defined because all but a finite number of \( a_i \) and \( b_i \) are zero.
Proposition: The inner product of two distinct prime numbers \( p_i \) and \( p_j \):
\( \langle p_i | p_j \rangle = \delta_{ij} \)
Proof: \( p_i \) serves a standard basis vector with 1 in the \( i \)-th position and zeros elsewhere. Similarly for \( p_j \). Then using (4) it is obvious that (5) is prooven and \( p_i \) and \( p_j \) are orthogonal.
Proposition: If \( n \) and \( m \) are coprime, then \( \langle n | m \rangle = 0 \).
Proof: Since \( n \) and \( m \) share no common prime factors, their exponent vectors have no overlapping non-zero entries. Therefore, the inner product sums over zeros:
\( \langle n | m \rangle = \sum_{i=1}^\infty a_i b_i = 0 \)
Proposition: If \( m \) divides \( n \), then \( \langle m | n \rangle = \sum_{i=1}^\infty a_i b_i \), where \( a_i \geq b_i \) for all \( i \).
Proof: Since \( m \) divides \( n \), for each prime \( p_i \), the exponent \( b_i \leq a_i \). The inner product sums the products of these exponents, reflecting the shared prime components.
Multiplication of numbers corresponds to the addition of their exponent vectors:
\( n \times m \mapsto [a_1 + b_1, a_2 + b_2, \ldots] \)
The inner product distributes accordingly, reflecting the additive nature of exponents in multiplication.
Examples
Try it yourself:
The norm of a number \( n \) is defined as:
\( \| n \| = \sqrt{\langle n | n \rangle} \)
Proposition: The norm of every prime is by definion 1.
Proof:\( \| p_i \| = \sqrt{\langle p_i | p_i \rangle} = \sqrt{1^2} = 1\)
Examples:
Interpretation
\( \| n \| \) is the Euclean norm or \( l^2\) norm. It can be interpreted as the Euclidean distance from the origin [0,0,0,..] in the \( \mathbb{N}^\infty \) factor space.
It also quantifies the spread or mass of a number across the prime dimension. A larger norm means many prime factors or a few large prime factors.
Try it yourself:
The angle \( \theta \) between two numbers \( n \) and \( m \) is given by:
\( \cos \theta = \frac{\langle n | m \rangle}{\| n \| \| m \|} \)
Proposition: The angle between any two distinct prime numbers is 90 degrees.
Proof: For distinct primes \( p_i \) and \( p_j \), \( \langle p_i | p_j \rangle = 0 \), and \( \| p_i \| = \| p_j \| = 1 \). Therefore according to definition (8): \( \cos \theta = 0 \implies \theta = \arccos(0) = \frac{\pi}{2} \text{ radians} \)
Proposition: The angle between any two coprime numbers is 90 degrees.
Proof: Coprime number share no prime factors, so for coprime \( n \) and \( m \) using (5), when \(a_i >0\) then \( b_i = 0\) and vice-versa. Therefore \( \langle n | m \rangle = \sum_{i=1}^\infty a_i b_i = 0 \) and using definition (8) \( \cos \theta = 0 \implies \theta = \arccos(0) = \frac{\pi}{2} \text{ radians} \)
We can calculate the angle between any two numbers using their exponent vectors.
Try it yourself:
The distance between two numbers \( n \) and \( m \) is the Euclidean distance between their exponent vectors:
\( d(n, m) = \sqrt{\sum_{i=1}^\infty (a_i - b_i)^2} \)
For practical calculations, since only a finite number of exponents are non-zero, the sum is finite.
Try it yourself:
We have introduced a geometric representation of numbers in the space \( \mathbb{N}^\infty \) defining an inner product, norm, distance, and angle to quantify number-theoretic relationships. This approach bridges number theory with geometry, offering new ways to analyze and classify numbers. Further work can explore deeper connections to algebraic number theory, combinatorial geometry, and computational methods.