\( D(4) = 64p + 16q + 4r + s = 84 \)

["# Unlocking Solutions to ( D(4) = 64p + 16q + 4r + s = 84 ): A Comprehensive Guide", "In the world of Diophantine equations—integer solutions bounded by linear combinations—equations of the form ( D(4) = 64p + 16q + 4r + s = 84 ) present a fascinating and structured challenge. This gauge function, combining coefficients with distinct powers of integers, defines a constrained integer relation with real-world implications in number theory, optimization, and coding systems.", "---", "## Understanding the Equation: ( D(4) = 64p + 16q + 4r + s = 84 )", "At its core, the equation\n[\n64p + 16q + 4r + s = 84\n]\nrepresents a total sum composed of four variables: ( p, q, r, ) and ( s ), each multiplied by increasingly smaller coefficients. Each term plays a pivotal role:", "- ( 64p ): weighted heavily by ( p ), showing strong influence on the total.\n- ( 16q ) contributes significantly but less dominantly than ( p ).\n- ( 4r ) moderates the contribution with medium weight.\n- ( s ), unweighted, acts as a residual term, ensuring flexibility to adjust solutions.", "This hierarchical structure allows for varied combinations that maintain integer or rational feasibility, depending on context.", "---", "## Key Objectives: Finding Integer Solutions", "Our goal is to identify integer (typically non-negative, positive) values of ( p, q, r, s ) that satisfy the equation. Since we’re solving a linear Diophantine equation in four variables, solutions often emerge through modular arithmetic, bounding techniques, and generating strategies.", "### Step 1: Modulo Reduction", "We analyze modulo the largest coefficient, 64:\n[\n16q + 4r + s \equiv 84 \pmod{64}\n]\nThen ( 84 \mod 64 = 20 ), so:\n[\n16q + 4r + s \equiv 20 \pmod{64}\n]", "Since ( 16q ) dominates this modulus, focus ( q ) to match the residue:\nTry ( q = 0, 1, ) or ( 2 ) (as ( 16 \cdot 3 = 48 > 20 ))", "- ( q = 0 \Rightarrow 4r + s \equiv 20 \pmod{64} )\n- ( q = 1 \Rightarrow 4r + s \equiv 4 \pmod{64} )\n- ( q = 2 \Rightarrow 4r + s \equiv -12 \equiv 52 \pmod{64} )", "These modular constraints sharply narrow possible values of ( q ) and help identify feasible ( r, s ).", "---", "## Step 2: Parameterizing with Reduced Variables", "Let us define:\n[\nT = 16q + 4r + s - 84 = 0\n]\nRewriting:\n[\ns = 84 - 64p - 16q - 4r\n]\nTo ensure ( s \in \mathbb{Z} ) and remains non-negative (if restricted to positives), bound ( p ). Since ( 64p \leq 84 \Rightarrow p \leq 1 ).", "Hence, only ( p = 0 ) or ( p = 1 ) are feasible.", "---", "### Case 1: ( p = 1 )", "Then:\n[\n64(1) + 16q + 4r + s = 84 \Rightarrow 16q + 4r + s = 20\n]\nWe want integer solutions in non-negative ( q, r, s ) satisfying ( 16q + 4r + s = 20 ).", "Try ( q = 0 ):\n[\n4r + s = 20 \Rightarrow s = 20 - 4r\n]\n( r ) can range from 0 to 5:", "- ( r = 0 ): ( s = 20 )\n- ( r = 1 ): ( s = 16 )\n- ( r = 2 ): ( s = 12 )\n- ( r = 3 ): ( s = 8 )\n- ( r = 4 ): ( s = 4 )\n- ( r = 5 ): ( s = 0 )", "All valid. Total solutions: 6.", "Try ( q = 1 ):\n[\n16(1) + 4r + s = 20 \Rightarrow 4r + s = 4 \Rightarrow s = 4 - 4r\n]\n( r = 0 ): ( s = 4 )\n( r = 1 ): ( s = 0 )\n( r \geq 2 \Rightarrow s < 0 ), invalid.", "Solutions: 2 more.", "( q \geq 2 \Rightarrow 16q \geq 32 > 20 ), infeasible.", "So total for ( p = 1 ): ( 6 + 2 = 8 ) integer solutions.", "---", "### Case 2: ( p = 0 )", "Then:\n[\n16q + 4r + s = 84\n]", "Again, modulo 16:\n[\n4r + s \equiv 84 \pmod{16}\n]\n( 84 \div 16 = 5 \ imes 16 = 80 \Rightarrow 84 \equiv 4 \pmod{16} )", "Thus:\n[\n4r + s \equiv 4 \pmod{16}\n]", "Let ( r = 0,1,\dots,3 ) (since ( 4 \cdot 4 = 16 > 4 )):", "- ( r = 0 ): ( s \equiv 4 \pmod{16} \Rightarrow s = 4 + 16k ), but ( s \geq 0 ). Minimum: ( s = 4 )\n- ( r = 1 ): ( s \equiv 0 \pmod{16} \Rightarrow s = 16k )\n- ( r = 2 ): ( s \equiv -4 \equiv 12 \pmod{16} \Rightarrow s = 12 + 16k )\n- ( r = 3 ): ( s \equiv -8 \equiv 8 \pmod{16} \Rightarrow s = 8 + 16k )", "Now find non-negative integer triples ( (q, r, s) ) for each residue class.", "For each fixed ( r ), solve ( 4r + s = 84 - 16q ), with ( s \geq 0 ), hence ( 16q \leq 84 \Rightarrow q \leq 5 )", "We iterate ( q = 0 ) to ( 5 ), checking feasibility for each residue.", "Example for ( r = 0, q = 5 ):\n( 16(5) = 80 ), so ( 4(0) + s = 4 \Rightarrow s = 4 ). Valid.", "Continue systematic counting:", "- ( q = 0 ): ( s = 84 - 16q - 4r = 84 - 4r )\n ( 84 - 4r \geq 0 \Rightarrow r \leq 21 ), but ( 4r \leq 84 \Rightarrow r \leq 21 ), but from modulo ( r \leq 3 ), so ( r = 0,1,2,3 ) → 4 solutions", "For each fixed ( q ), for each residue ( r \mod 16 ), check if ( s = 84 - 16q - 4r \geq 0 ) and satisfies congruence.", "Since for each ( q ), exactly one residue ( r \mod 16 ) works, and ( r \geq 0 ), and ( 4r \leq 84 - 16q ), count valid ( r ) from 0 upward such that ( 4r \leq 84 - 16q ).", "- ( q = 0 ): ( 4r \leq 84 \Rightarrow r \leq 21 ), but ( r \equiv 4 \pmod{16} \Rightarrow r = 0, 16 ) → 2 values\n- ( q = 1 ): ( 4r \leq 68 \Rightarrow r \leq 17 ), ( r \equiv 4 \pmod{16} \Rightarrow r = 0, 16 ) → valid: ( r = 0 ) (16×0=0 ≤ 68), ( r = 16 ) (64 ≤ 68) → 2\n- ( q = 2 ): ( 4r \leq 52 \Rightarrow r \leq 13 ), ( r \equiv 4 \pmod{16} \Rightarrow r = 0, 16 ) → only ( r = 0 ) → 1\n- ( q = 3 ): ( 4r \leq 36 \Rightarrow r \leq 9 ), only ( r = 0 ) → 1\n- ( q = 4 ): ( 4r \leq 20 \Rightarrow r \leq 5 ), ( r = 0 ) → 1\n- ( q = 5 ): ( 4r \leq 4 \Rightarrow r \leq 1 ), ( r \equiv 4 \pmod{16} \Rightarrow r = 0 ) only → 1\n- ( q \geq 6 ): ( 16q \geq 96 > 84 ) → invalid", "Total for ( p = 0 ): ( 2 + 2 + 1 + 1 + 1 + 1 = 8 ) solutions.", "---", "## Summary of Total Solutions", "| Case ( p ) | Number of Solutions |\n|-------------|--------------------|\n| ( p = 1 ) | 8 |\n| ( p = 0 ) | 8 |\n| Total | 16 |", "All solutions have ( s \geq 0 ) and are integers. No negative variables assumed; if restricted to positive integers, exclude ( s = 0 ), reducing count slightly.", "---", "## Practical Applications and Significance", "Equations like ( D(4) = 64p + 16q + 4r + s = 84 ) model constrained resource allocation, where:", "- ( p, q, r ) represent scaled measures (e.g., batches, yards, volts)\n- ( s ) acts as leftover or residual\n- Coefficients reflect differing scaling weights (power-of-two descent)", "Such forms appear in:", "- Coding theory — error-correcting codes with digit constraints\n- Cryptography — modular structures with lattice-like variables\n- Optimization — integer programming with hierarchical variable priorities\n- Number theory* — representation of integers with valuations", "---", "## Conclusion: Mastering Structured Diophantine Forms", "The equation ( 64p + 16q + 4r + s = 84 ) exemplifies how hierarchical linear forms generate ordered integer solutions. By combining modular arithmetic, parameter bounding, and variable isolation, we decode solution sets efficiently.", "Whether for theoretical exploration or applied modeling, understanding such Diophantine equations sharpens analytical reasoning and unlocks deeper insights into discrete mathematics.", "---", "### Keywords:\nDiophantine equation, ( D(4) = 64p + 16q + 4r + s = 84 ), integer solutions, modular arithmetic, bounded variables, number theory, optimization. \n---", "Explore more: For deeper dives into lattice variable systems and Diophantine parametrization, examine modular reduction strategies and integer lattice point enumeration in computational number theory."]