garywelz/programming_framework
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1<!DOCTYPE html>2<html lang="en">3<head>4 <meta charset="UTF-8" />5 <meta name="viewport" content="width=device-width, initial-scale=1" />6 <title>Mathematics Batch 01 - Number Theory - Programming Framework Analysis</title>7 <style>8 body { 9 font-family: 'Times New Roman', Times, serif, 'Arial Unicode MS'; 10 margin: 0; 11 background: #ffffff; 12 color: #000000; 13 line-height: 1.6; 14 font-size: 12pt; 15 }16 .container { 17 max-width: 1000px; 18 margin: 0 auto; 19 padding: 1.5rem; 20 }21 h1, h2, h3 { 22 color: #000000; 23 margin-top: 1.5rem; 24 margin-bottom: 0.75rem; 25 }26 h1 { 27 font-size: 18pt; 28 text-align: center; 29 }30 h2 { 31 font-size: 16pt; 32 border-bottom: 2px solid #000; 33 padding-bottom: 0.5rem; 34 }35 h3 { 36 font-size: 14pt; 37 }38 p { 39 margin-bottom: 1rem; 40 text-align: justify; 41 }42 .figure { 43 margin: 2rem 0; 44 text-align: center; 45 border: 1px solid #ccc; 46 padding: 1rem; 47 background: #f9f9f9; 48 }49 .figure-caption { 50 margin-top: 1rem; 51 font-style: italic; 52 text-align: left; 53 }54 .mermaid { 55 background: white; 56 padding: 1rem; 57 border-radius: 4px; 58 }59 .navigation {60 margin: 3rem 0;61 padding: 1rem;62 background: #f8f9fa;63 border-radius: 8px;64 }65 .nav-links {66 display: flex;67 flex-wrap: wrap;68 gap: 1rem;69 justify-content: center;70 }71 .nav-link {72 color: #007bff;73 text-decoration: none;74 padding: 0.5rem 1rem;75 border: 1px solid #007bff;76 border-radius: 4px;77 transition: all 0.3s ease;78 }79 .nav-link:hover {80 background: #007bff;81 color: white;82 }83 .footer {84 margin-top: 3rem;85 padding: 1rem;86 background: #f8f9fa;87 border-radius: 8px;88 text-align: center;89 }90 .contact-info {91 margin-top: 1rem;92 }93 .contact-info p {94 margin: 0.25rem 0;95 text-align: center;96 }97 </style>98 <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>99 <script>100 mermaid.initialize({ 101 startOnLoad: true, 102 theme: 'default', 103 flowchart: { 104 useMaxWidth: false, 105 htmlLabels: true,106 curve: 'linear',107 nodeSpacing: 30,108 rankSpacing: 30,109 padding: 10110 },111 themeVariables: {112 fontFamily: 'Arial Unicode MS, Arial, sans-serif'113 }114 });115 </script>116</head>117<body>118 <div class="container">119 <h1>Mathematics Batch 01 - Number Theory - Programming Framework Analysis</h1>120 121 <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>122 123 <h2>1. Prime Number Generation Process</h2>124 <div class="figure">125 <div class="mermaid">126graph TD127 A1[Range Definition] --> B1[Sieve Algorithm Selection]128 C1[Upper Bound] --> D1[Memory Allocation]129 E1[Optimization Strategy] --> F1[Algorithm Implementation]130 131 B1 --> G1[Eratosthenes Sieve]132 D1 --> H1[Boolean Array Creation]133 F1 --> I1[Marking Strategy]134 135 G1 --> J1[Initialize All True]136 H1 --> K1[Array Size Calculation]137 I1 --> L1[Prime Detection]138 139 J1 --> M1[Mark 0 and 1 False]140 K1 --> L1141 L1 --> N1[Square Root Optimization]142 143 M1 --> O1[Iterate from 2]144 N1 --> P1[Stop at Square Root]145 O1 --> Q1[Prime Number Process]146 147 P1 --> R1[Mark Multiples False]148 Q1 --> S1[Collect Prime Numbers]149 R1 --> T1[Prime Number Result]150 151 S1 --> U1[Prime Number Validation]152 T1 --> V1[Prime Number Count]153 U1 --> W1[Prime Number Output]154 155 V1 --> X1[Prime Number Analysis]156 W1 --> Y1[Prime Number Final Result]157 X1 --> Z1[Prime Number Analysis Complete]158 159 style A1 fill:#ff6b6b,color:#fff160 style C1 fill:#ff6b6b,color:#fff161 style E1 fill:#ff6b6b,color:#fff162 163 style B1 fill:#ffd43b,color:#000164 style D1 fill:#ffd43b,color:#000165 style F1 fill:#ffd43b,color:#000166 style G1 fill:#ffd43b,color:#000167 style H1 fill:#ffd43b,color:#000168 style I1 fill:#ffd43b,color:#000169 style J1 fill:#ffd43b,color:#000170 style K1 fill:#ffd43b,color:#000171 style L1 fill:#ffd43b,color:#000172 style M1 fill:#ffd43b,color:#000173 style N1 fill:#ffd43b,color:#000174 style O1 fill:#ffd43b,color:#000175 style P1 fill:#ffd43b,color:#000176 style Q1 fill:#ffd43b,color:#000177 style R1 fill:#ffd43b,color:#000178 style S1 fill:#ffd43b,color:#000179 style T1 fill:#ffd43b,color:#000180 style U1 fill:#ffd43b,color:#000181 style V1 fill:#ffd43b,color:#000182 style W1 fill:#ffd43b,color:#000183 style X1 fill:#ffd43b,color:#000184 style Y1 fill:#ffd43b,color:#000185 style Z1 fill:#ffd43b,color:#000186 187 style M1 fill:#51cf66,color:#fff188 style N1 fill:#51cf66,color:#fff189 style O1 fill:#51cf66,color:#fff190 style P1 fill:#51cf66,color:#fff191 style Q1 fill:#51cf66,color:#fff192 style R1 fill:#51cf66,color:#fff193 style S1 fill:#51cf66,color:#fff194 style T1 fill:#51cf66,color:#fff195 style U1 fill:#51cf66,color:#fff196 style V1 fill:#51cf66,color:#fff197 style W1 fill:#51cf66,color:#fff198 style X1 fill:#51cf66,color:#fff199 style Y1 fill:#51cf66,color:#fff200 style Z1 fill:#51cf66,color:#fff201 202 style Z1 fill:#b197fc,color:#fff203 </div>204 205 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">206 <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;">207 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs208 </div>209 <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;">210 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Number Theory Methods211 </div>212 <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;">213 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Prime Operations214 </div>215 <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;">216 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates217 </div>218 <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;">219 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products220 </div>221 </div>222 223 <div class="figure-caption">224 <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.225 </div>226 </div>227 228 <h2>2. Modular Arithmetic Process</h2>229 <div class="figure">230 <div class="mermaid">231graph TD232 A2[Integer a] --> B2[Modulus m]233 C2[Operation Type] --> D2[Modular Arithmetic]234 E2[Congruence Analysis] --> F2[Residue Calculation]235 236 B2 --> G2[Modulo Operation]237 D2 --> H2[Addition Modulo m]238 F2 --> I2[Multiplication Modulo m]239 240 G2 --> J2[Division Modulo m]241 H2 --> K2[Subtraction Modulo m]242 I2 --> L2[Exponentiation Modulo m]243 244 J2 --> M2[Multiplicative Inverse]245 K2 --> L2246 L2 --> N2[Fermat Little Theorem]247 248 M2 --> O2[Extended Euclidean Algorithm]249 N2 --> P2[Euler Totient Function]250 O2 --> Q2[Modular Arithmetic Process]251 252 P2 --> R2[Chinese Remainder Theorem]253 Q2 --> S2[Modular Arithmetic Validation]254 R2 --> T2[Modular Arithmetic Result]255 256 S2 --> U2[Modular Arithmetic Analysis]257 T2 --> V2[Modular Arithmetic Parameters]258 U2 --> W2[Modular Arithmetic Output]259 260 V2 --> X2[Modular Arithmetic Analysis]261 W2 --> Y2[Modular Arithmetic Final Result]262 X2 --> Z2[Modular Arithmetic Analysis Complete]263 264 style A2 fill:#ff6b6b,color:#fff265 style C2 fill:#ff6b6b,color:#fff266 style E2 fill:#ff6b6b,color:#fff267 268 style B2 fill:#ffd43b,color:#000269 style D2 fill:#ffd43b,color:#000270 style F2 fill:#ffd43b,color:#000271 style G2 fill:#ffd43b,color:#000272 style H2 fill:#ffd43b,color:#000273 style I2 fill:#ffd43b,color:#000274 style J2 fill:#ffd43b,color:#000275 style K2 fill:#ffd43b,color:#000276 style L2 fill:#ffd43b,color:#000277 style M2 fill:#ffd43b,color:#000278 style N2 fill:#ffd43b,color:#000279 style O2 fill:#ffd43b,color:#000280 style P2 fill:#ffd43b,color:#000281 style Q2 fill:#ffd43b,color:#000282 style R2 fill:#ffd43b,color:#000283 style S2 fill:#ffd43b,color:#000284 style T2 fill:#ffd43b,color:#000285 style U2 fill:#ffd43b,color:#000286 style V2 fill:#ffd43b,color:#000287 style W2 fill:#ffd43b,color:#000288 style X2 fill:#ffd43b,color:#000289 style Y2 fill:#ffd43b,color:#000290 style Z2 fill:#ffd43b,color:#000291 292 style M2 fill:#51cf66,color:#fff293 style N2 fill:#51cf66,color:#fff294 style O2 fill:#51cf66,color:#fff295 style P2 fill:#51cf66,color:#fff296 style Q2 fill:#51cf66,color:#fff297 style R2 fill:#51cf66,color:#fff298 style S2 fill:#51cf66,color:#fff299 style T2 fill:#51cf66,color:#fff300 style U2 fill:#51cf66,color:#fff301 style V2 fill:#51cf66,color:#fff302 style W2 fill:#51cf66,color:#fff303 style X2 fill:#51cf66,color:#fff304 style Y2 fill:#51cf66,color:#fff305 style Z2 fill:#51cf66,color:#fff306 307 style Z2 fill:#b197fc,color:#fff308 </div>309 310 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">311 <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;">312 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs313 </div>314 <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;">315 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Modular Methods316 </div>317 <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;">318 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Modular Operations319 </div>320 <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;">321 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates322 </div>323 <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;">324 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products325 </div>326 </div>327 328 <div class="figure-caption">329 <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.330 </div>331 </div>332 333 <h2>3. Diophantine Equations Process</h2>334 <div class="figure">335 <div class="mermaid">336graph TD337 A3[Linear Equation] --> B3[Variable Analysis]338 C3[Coefficient Analysis] --> D3[Solution Strategy]339 E3[Integer Constraints] --> F3[Existence Check]340 341 B3 --> G3[Two Variable Equation]342 D3 --> H3[Extended Euclidean Algorithm]343 F3 --> I3[GCD Analysis]344 345 G3 --> J3[General Solution]346 H3 --> K3[Particular Solution]347 I3 --> L3[Parametric Form]348 349 J3 --> M3[Solution Verification]350 K3 --> L3351 L3 --> N3[Solution Validation]352 353 M3 --> O3[Integer Solutions]354 N3 --> P3[Solution Bounds]355 O3 --> Q3[Diophantine Equations Process]356 357 P3 --> R3[Solution Enumeration]358 Q3 --> S3[Diophantine Equations Validation]359 R3 --> T3[Diophantine Equations Result]360 361 S3 --> U3[Diophantine Equations Analysis]362 T3 --> V3[Diophantine Equations Parameters]363 U3 --> W3[Diophantine Equations Output]364 365 V3 --> X3[Diophantine Equations Analysis]366 W3 --> Y3[Diophantine Equations Final Result]367 X3 --> Z3[Diophantine Equations Analysis Complete]368 369 style A3 fill:#ff6b6b,color:#fff370 style C3 fill:#ff6b6b,color:#fff371 style E3 fill:#ff6b6b,color:#fff372 373 style B3 fill:#ffd43b,color:#000374 style D3 fill:#ffd43b,color:#000375 style F3 fill:#ffd43b,color:#000376 style G3 fill:#ffd43b,color:#000377 style H3 fill:#ffd43b,color:#000378 style I3 fill:#ffd43b,color:#000379 style J3 fill:#ffd43b,color:#000380 style K3 fill:#ffd43b,color:#000381 style L3 fill:#ffd43b,color:#000382 style M3 fill:#ffd43b,color:#000383 style N3 fill:#ffd43b,color:#000384 style O3 fill:#ffd43b,color:#000385 style P3 fill:#ffd43b,color:#000386 style Q3 fill:#ffd43b,color:#000387 style R3 fill:#ffd43b,color:#000388 style S3 fill:#ffd43b,color:#000389 style T3 fill:#ffd43b,color:#000390 style U3 fill:#ffd43b,color:#000391 style V3 fill:#ffd43b,color:#000392 style W3 fill:#ffd43b,color:#000393 style X3 fill:#ffd43b,color:#000394 style Y3 fill:#ffd43b,color:#000395 style Z3 fill:#ffd43b,color:#000396 397 style M3 fill:#51cf66,color:#fff398 style N3 fill:#51cf66,color:#fff399 style O3 fill:#51cf66,color:#fff400 style P3 fill:#51cf66,color:#fff401 style Q3 fill:#51cf66,color:#fff402 style R3 fill:#51cf66,color:#fff403 style S3 fill:#51cf66,color:#fff404 style T3 fill:#51cf66,color:#fff405 style U3 fill:#51cf66,color:#fff406 style V3 fill:#51cf66,color:#fff407 style W3 fill:#51cf66,color:#fff408 style X3 fill:#51cf66,color:#fff409 style Y3 fill:#51cf66,color:#fff410 style Z3 fill:#51cf66,color:#fff411 412 style Z3 fill:#b197fc,color:#fff413 </div>414 415 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">416 <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;">417 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs418 </div>419 <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;">420 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Diophantine Methods421 </div>422 <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;">423 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Solution Operations424 </div>425 <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;">426 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates427 </div>428 <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;">429 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products430 </div>431 </div>432 433 <div class="figure-caption">434 <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.435 </div>436 </div>437 438 <div class="navigation">439 <h3>Navigation</h3>440 <div class="nav-links">441 <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a>442 <a href="mathematics_batch_02.html" class="nav-link">Next: Analysis & Calculus →</a>443 <a href="index.html" class="nav-link">Programming Framework Home</a>444 </div>445 </div>446 447 <div class="footer">448 <p><strong>Generated using the Programming Framework methodology</strong></p>449 <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>450 <div class="contact-info">451 <p><strong>Gary Welz</strong></p>452 <p>Retired Faculty Member</p>453 <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p>454 <p>Borough of Manhattan Community College, CUNY</p>455 <p>CUNY Graduate Center (New Media Lab)</p>456 <p>Email: gwelz@jjay.cuny.edu</p>457 </div>458 </div>459 </div>460</body>461</html>462