Term: b-adic Power (P_b(N))
Definition: The "body" of an integer N relative to a chosen base b. It is the largest positive divisor of N whose prime factors are all also prime factors of the base b.
Chapter 1: The Native Animals (Elementary School Understanding)
Let's go back to our special countries where numbers live. Each country has its own "native" prime animals.
Base-10 Land: The native animals are 2s and 5s.
Base-6 Land: The native animals are 2s and 3s.
The b-adic Power is the part of a number that is a "native" to that specific land. It's the herd of all the animals in a number's recipe that belong in that country.
Let's look at the number 72. Its secret recipe is 2×2×2 × 3×3.
In Base-10 Land, the "2" animals are native, but the "3" animals are foreigners. So, the native part, the P₁₀(72), is the herd of all the 2s: 2×2×2 = 8.
In Base-6 Land, both "2" and "3" animals are native. So, the native part, P₆(72), is the entire herd! 2×2×2 × 3×3 = 72.
In Base-7 Land, both "2" and "3" are foreigners. The number 7 is the only native. There are no native animals in the recipe for 72. So, the native part, P₇(72), is just 1 (the placeholder for an empty herd).
The b-adic Power is the piece of a number's recipe that perfectly matches the recipe of the base. It's the number's "native body" as seen from that specific country.
Chapter 2: The Part That Belongs (Middle School Understanding)
The b-adic Power, P_b(N), is the part of an integer N that is "structurally native" to the base b. It is built exclusively from the prime factors that N and b share.
The Rule: P_b(N) is the largest divisor of N whose prime factors can all be found in the prime factorization of b.
How to Find It:
Find the prime factors of your base b. Let's call them the "native primes."
Take your number N and find all of its prime factors.
The b-adic Power is the product of all the prime powers in N's factorization whose base prime is one of the "native primes."
Example: Find the Power of N=300 for different bases.
The prime factorization of 300 is 2² × 3 × 5².
For base b=10:
The native primes are {2, 5}.
We look at the prime factors of 300 and pick out all the powers of 2s and 5s.
The native part is 2² × 5² = 4 × 25 = 100.
So, P₁₀(300) = 100.
For base b=6:
The native primes are {2, 3}.
We look at the prime factors of 300 and pick out all the powers of 2s and 3s.
The native part is 2² × 3¹ = 4 × 3 = 12.
So, P₆(300) = 12.
The Power changes depending on the base you are using as your "point of view." It represents the part of the number that is structurally "at home" in that base. The most commonly used one is the Dyadic Power (P₂), which is just the number's largest power-of-two divisor.
Chapter 3: The Base-Native Part of the Soul (High School Understanding)
The b-adic Power, P_b(N), is a formal component derived from the b-adic Decomposition (N = P_b(N) × K_b(N)).
Formal Definition: The b-adic Power P_b(N) is the unique positive integer such that N = P_b(N) × K_b(N), where K_b(N) is the largest divisor of N that is coprime to b.
This means that P_b(N) is composed exclusively of prime factors that are also prime factors of b.
Relationship to the Algebraic Soul:
The Algebraic Soul is the complete prime factorization of N. The P_b(N) is a specific subset of that soul. It is the product of all the prime power factors in N's soul whose base primes are also in the soul of b.
Example: N = 396, b = 30
Find the Souls:
Soul of N = 396 is 2² × 3² × 11.
Soul of b = 30 is 2 × 3 × 5.
Identify the "Native" Primes: The prime factors that both N and b share are {2, 3}.
Construct the Power: The Power is the product of all the powers of these native primes as they appear in N.
P₃₀(396) = 2² × 3² = 36.
The Power is, by convention, always positive, even if N is negative. This forces the Kernel to carry the sign.
Connection to the Arithmetic Body:
The P_b(N) is the component that has the simplest possible representation in base b. For example, in base-10, P₁₀(300) = 100, which is a simple 1 followed by zeros. In base-2, P₂(12) = 4, which is 100₂. The b-adic Power represents the "structurally simple" part of N relative to the base b.
Chapter 4: A Projection onto a Sub-Monoid (College Level)
The b-adic Power, P_b(N), is the result of a projection of the integer N onto the multiplicative monoid generated by the prime factors of the base b.
Formal Definition via p-adic Valuations:
Let rad(b) be the radical of b (the set of its distinct prime factors). The b-adic Power is defined as:
P_b(N) = Π_{p | p ∈ rad(b)} [ p^v_p(|N|) ]
where v_p(|N|) is the p-adic valuation of the absolute value of N for each prime p that is also a prime factor of b.
Significance in Structural Dynamics:
The b-adic Power is the component of a number's structure that is "transparent" or "harmonious" with the base b. When a number N is represented in base b, the structural complexity of its Arithmetic Body arises almost entirely from its b-adic Kernel, K_b(N). The P_b(N) component contributes a simple, predictable pattern.
Example: N = 300 in base-10. N = K₁₀(300) × P₁₀(300) = 3 × 100.
The representation of P₁₀(300)=100 is 100₁₀. It is simple.
The representation of K₁₀(300)=3 is 3₁₀.
The representation of 300 is simply the representation of the Kernel, shifted by the magnitude of the Power.
This allows the framework to isolate the source of Frame Incompatibility. The difficulty of representing N in base b is equivalent to the difficulty of representing K_b(N) in base b. The P_b(N) part is structurally trivial within that frame.
Chapter 5: Worksheet - Finding the Native
Part 1: The Native Animals (Elementary Level)
In "Base-10 Land" (native animals 2 and 5), what is the "Native Herd" part (Power) of the number 90 (2 × 3² × 5)?
What is the "Native Herd" part of 90 in "Base-21 Land" (native animals 3 and 7)?
Part 2: The Part That Belongs (Middle School Level)
What does it mean for a number's prime factors to be "native" to a base b?
Find P₁₀(175). (Prime factors of 10 are {2, 5}; 175 = 5² × 7).
Find P₇(175).
Part 3: The Base-Native Soul (High School Level)
The prime factorization of N is 2⁴ × 3¹ × 5³. The base is b=20.
What is the Algebraic Soul of b?
Identify the "native" prime powers in N's soul.
Calculate P₂₀(N).
What is the b-adic Power of any number N if the base b is a prime number that does not divide N?
Part 4: Projections (College Level)
What is the radical of a number b? What is rad(72)?
If two bases b₁ and b₂ are commensurable, what does this imply about their radicals and, consequently, about their b-adic Powers for any given N?
Explain the statement: "The P_b(N) component is structurally transparent in base b."