Mathematics Batch 01 - Number Theory - Programming Framework Analysis
This document presents number theory processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.
1. Prime Number Generation Process
graph TD
A1[Range Definition] --> B1[Sieve Algorithm Selection]
C1[Upper Bound] --> D1[Memory Allocation]
E1[Optimization Strategy] --> F1[Algorithm Implementation]
B1 --> G1[Eratosthenes Sieve]
D1 --> H1[Boolean Array Creation]
F1 --> I1[Marking Strategy]
G1 --> J1[Initialize All True]
H1 --> K1[Array Size Calculation]
I1 --> L1[Prime Detection]
J1 --> M1[Mark 0 and 1 False]
K1 --> L1
L1 --> N1[Square Root Optimization]
M1 --> O1[Iterate from 2]
N1 --> P1[Stop at Square Root]
O1 --> Q1[Prime Number Process]
P1 --> R1[Mark Multiples False]
Q1 --> S1[Collect Prime Numbers]
R1 --> T1[Prime Number Result]
S1 --> U1[Prime Number Validation]
T1 --> V1[Prime Number Count]
U1 --> W1[Prime Number Output]
V1 --> X1[Prime Number Analysis]
W1 --> Y1[Prime Number Final Result]
X1 --> Z1[Prime Number Analysis Complete]
style A1 fill:#ff6b6b,color:#fff
style C1 fill:#ff6b6b,color:#fff
style E1 fill:#ff6b6b,color:#fff
style B1 fill:#ffd43b,color:#000
style D1 fill:#ffd43b,color:#000
style F1 fill:#ffd43b,color:#000
style G1 fill:#ffd43b,color:#000
style H1 fill:#ffd43b,color:#000
style I1 fill:#ffd43b,color:#000
style J1 fill:#ffd43b,color:#000
style K1 fill:#ffd43b,color:#000
style L1 fill:#ffd43b,color:#000
style M1 fill:#ffd43b,color:#000
style N1 fill:#ffd43b,color:#000
style O1 fill:#ffd43b,color:#000
style P1 fill:#ffd43b,color:#000
style Q1 fill:#ffd43b,color:#000
style R1 fill:#ffd43b,color:#000
style S1 fill:#ffd43b,color:#000
style T1 fill:#ffd43b,color:#000
style U1 fill:#ffd43b,color:#000
style V1 fill:#ffd43b,color:#000
style W1 fill:#ffd43b,color:#000
style X1 fill:#ffd43b,color:#000
style Y1 fill:#ffd43b,color:#000
style Z1 fill:#ffd43b,color:#000
style M1 fill:#51cf66,color:#fff
style N1 fill:#51cf66,color:#fff
style O1 fill:#51cf66,color:#fff
style P1 fill:#51cf66,color:#fff
style Q1 fill:#51cf66,color:#fff
style R1 fill:#51cf66,color:#fff
style S1 fill:#51cf66,color:#fff
style T1 fill:#51cf66,color:#fff
style U1 fill:#51cf66,color:#fff
style V1 fill:#51cf66,color:#fff
style W1 fill:#51cf66,color:#fff
style X1 fill:#51cf66,color:#fff
style Y1 fill:#51cf66,color:#fff
style Z1 fill:#51cf66,color:#fff
style Z1 fill:#b197fc,color:#fff
Triggers & Inputs
Number Theory Methods
Prime Operations
Intermediates
Products
Figure 1. Prime Number Generation Process. This number theory process visualization demonstrates the Sieve of Eratosthenes algorithm. The flowchart shows range inputs and optimization strategies, number theory methods and sieve algorithms, prime operations and marking strategies, intermediate results, and final prime number outputs.
2. Modular Arithmetic Process
graph TD
A2[Integer a] --> B2[Modulus m]
C2[Operation Type] --> D2[Modular Arithmetic]
E2[Congruence Analysis] --> F2[Residue Calculation]
B2 --> G2[Modulo Operation]
D2 --> H2[Addition Modulo m]
F2 --> I2[Multiplication Modulo m]
G2 --> J2[Division Modulo m]
H2 --> K2[Subtraction Modulo m]
I2 --> L2[Exponentiation Modulo m]
J2 --> M2[Multiplicative Inverse]
K2 --> L2
L2 --> N2[Fermat Little Theorem]
M2 --> O2[Extended Euclidean Algorithm]
N2 --> P2[Euler Totient Function]
O2 --> Q2[Modular Arithmetic Process]
P2 --> R2[Chinese Remainder Theorem]
Q2 --> S2[Modular Arithmetic Validation]
R2 --> T2[Modular Arithmetic Result]
S2 --> U2[Modular Arithmetic Analysis]
T2 --> V2[Modular Arithmetic Parameters]
U2 --> W2[Modular Arithmetic Output]
V2 --> X2[Modular Arithmetic Analysis]
W2 --> Y2[Modular Arithmetic Final Result]
X2 --> Z2[Modular Arithmetic Analysis Complete]
style A2 fill:#ff6b6b,color:#fff
style C2 fill:#ff6b6b,color:#fff
style E2 fill:#ff6b6b,color:#fff
style B2 fill:#ffd43b,color:#000
style D2 fill:#ffd43b,color:#000
style F2 fill:#ffd43b,color:#000
style G2 fill:#ffd43b,color:#000
style H2 fill:#ffd43b,color:#000
style I2 fill:#ffd43b,color:#000
style J2 fill:#ffd43b,color:#000
style K2 fill:#ffd43b,color:#000
style L2 fill:#ffd43b,color:#000
style M2 fill:#ffd43b,color:#000
style N2 fill:#ffd43b,color:#000
style O2 fill:#ffd43b,color:#000
style P2 fill:#ffd43b,color:#000
style Q2 fill:#ffd43b,color:#000
style R2 fill:#ffd43b,color:#000
style S2 fill:#ffd43b,color:#000
style T2 fill:#ffd43b,color:#000
style U2 fill:#ffd43b,color:#000
style V2 fill:#ffd43b,color:#000
style W2 fill:#ffd43b,color:#000
style X2 fill:#ffd43b,color:#000
style Y2 fill:#ffd43b,color:#000
style Z2 fill:#ffd43b,color:#000
style M2 fill:#51cf66,color:#fff
style N2 fill:#51cf66,color:#fff
style O2 fill:#51cf66,color:#fff
style P2 fill:#51cf66,color:#fff
style Q2 fill:#51cf66,color:#fff
style R2 fill:#51cf66,color:#fff
style S2 fill:#51cf66,color:#fff
style T2 fill:#51cf66,color:#fff
style U2 fill:#51cf66,color:#fff
style V2 fill:#51cf66,color:#fff
style W2 fill:#51cf66,color:#fff
style X2 fill:#51cf66,color:#fff
style Y2 fill:#51cf66,color:#fff
style Z2 fill:#51cf66,color:#fff
style Z2 fill:#b197fc,color:#fff
Triggers & Inputs
Modular Methods
Modular Operations
Intermediates
Products
Figure 2. Modular Arithmetic Process. This number theory process visualization demonstrates modular arithmetic operations and congruence analysis. The flowchart shows integer inputs and operation types, modular methods and arithmetic operations, modular operations and residue calculations, intermediate results, and final modular arithmetic outputs.
3. Diophantine Equations Process
graph TD
A3[Linear Equation] --> B3[Variable Analysis]
C3[Coefficient Analysis] --> D3[Solution Strategy]
E3[Integer Constraints] --> F3[Existence Check]
B3 --> G3[Two Variable Equation]
D3 --> H3[Extended Euclidean Algorithm]
F3 --> I3[GCD Analysis]
G3 --> J3[General Solution]
H3 --> K3[Particular Solution]
I3 --> L3[Parametric Form]
J3 --> M3[Solution Verification]
K3 --> L3
L3 --> N3[Solution Validation]
M3 --> O3[Integer Solutions]
N3 --> P3[Solution Bounds]
O3 --> Q3[Diophantine Equations Process]
P3 --> R3[Solution Enumeration]
Q3 --> S3[Diophantine Equations Validation]
R3 --> T3[Diophantine Equations Result]
S3 --> U3[Diophantine Equations Analysis]
T3 --> V3[Diophantine Equations Parameters]
U3 --> W3[Diophantine Equations Output]
V3 --> X3[Diophantine Equations Analysis]
W3 --> Y3[Diophantine Equations Final Result]
X3 --> Z3[Diophantine Equations Analysis Complete]
style A3 fill:#ff6b6b,color:#fff
style C3 fill:#ff6b6b,color:#fff
style E3 fill:#ff6b6b,color:#fff
style B3 fill:#ffd43b,color:#000
style D3 fill:#ffd43b,color:#000
style F3 fill:#ffd43b,color:#000
style G3 fill:#ffd43b,color:#000
style H3 fill:#ffd43b,color:#000
style I3 fill:#ffd43b,color:#000
style J3 fill:#ffd43b,color:#000
style K3 fill:#ffd43b,color:#000
style L3 fill:#ffd43b,color:#000
style M3 fill:#ffd43b,color:#000
style N3 fill:#ffd43b,color:#000
style O3 fill:#ffd43b,color:#000
style P3 fill:#ffd43b,color:#000
style Q3 fill:#ffd43b,color:#000
style R3 fill:#ffd43b,color:#000
style S3 fill:#ffd43b,color:#000
style T3 fill:#ffd43b,color:#000
style U3 fill:#ffd43b,color:#000
style V3 fill:#ffd43b,color:#000
style W3 fill:#ffd43b,color:#000
style X3 fill:#ffd43b,color:#000
style Y3 fill:#ffd43b,color:#000
style Z3 fill:#ffd43b,color:#000
style M3 fill:#51cf66,color:#fff
style N3 fill:#51cf66,color:#fff
style O3 fill:#51cf66,color:#fff
style P3 fill:#51cf66,color:#fff
style Q3 fill:#51cf66,color:#fff
style R3 fill:#51cf66,color:#fff
style S3 fill:#51cf66,color:#fff
style T3 fill:#51cf66,color:#fff
style U3 fill:#51cf66,color:#fff
style V3 fill:#51cf66,color:#fff
style W3 fill:#51cf66,color:#fff
style X3 fill:#51cf66,color:#fff
style Y3 fill:#51cf66,color:#fff
style Z3 fill:#51cf66,color:#fff
style Z3 fill:#b197fc,color:#fff
Triggers & Inputs
Diophantine Methods
Solution Operations
Intermediates
Products
Figure 3. Diophantine Equations Process. This number theory process visualization demonstrates integer solution finding for linear equations. The flowchart shows equation inputs and coefficient analysis, Diophantine methods and solution strategies, solution operations and verification, intermediate results, and final Diophantine equation outputs.