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| <title>Mathematics Batch 01 - Number Theory - Programming Framework Analysis</title> | |
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| <h1>Mathematics Batch 01 - Number Theory - Programming Framework Analysis</h1> | |
| <p>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.</p> | |
| <h2>1. Prime Number Generation Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Number Theory Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Prime Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 1.</strong> 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. | |
| </div> | |
| </div> | |
| <h2>2. Modular Arithmetic Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
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| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Modular Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Modular Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 2.</strong> 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. | |
| </div> | |
| </div> | |
| <h2>3. Diophantine Equations Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Diophantine Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Solution Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 3.</strong> 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. | |
| </div> | |
| </div> | |
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| <h3>Navigation</h3> | |
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| <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a> | |
| <a href="mathematics_batch_02.html" class="nav-link">Next: Analysis & Calculus →</a> | |
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| <div class="footer"> | |
| <p><strong>Generated using the Programming Framework methodology</strong></p> | |
| <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p> | |
| <div class="contact-info"> | |
| <p><strong>Gary Welz</strong></p> | |
| <p>Retired Faculty Member</p> | |
| <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p> | |
| <p>Borough of Manhattan Community College, CUNY</p> | |
| <p>CUNY Graduate Center (New Media Lab)</p> | |
| <p>Email: gwelz@jjay.cuny.edu</p> | |
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