Diophantine equations, specifically in the form x² - 4y² = n, require finding non-negative integer solutions for given positive integers n. The article presents an innovative method by rewriting the equation to (x - 2y)(x + 2y) = n and optimizing solution search through factor pairs of n. Focusing only on pairs where the same parity and divisibility conditions hold allows for accurate integer value calculations for x and y. The method significantly reduces complexity while yielding correct results, making it a valuable approach for number theory enthusiasts and functional programming applications.
The polynomial equation x² - 4y² = n seeks integer solutions for non-negative integers x and y, exploring optimized methods through factor pairs of n.
By rewriting the equation to (x - 2y)(x + 2y) = n, we can optimize our search for solutions by only considering factor pairs of n.
Ensuring conditions such as the same parity and divisibility by 4 for factor pairs of n is crucial before deriving integer values for x and y.
This approach combines number theory with efficient functional programming techniques, yielding results with a reduced computational complexity while ensuring correctness.
#diophantine-equations #number-theory #functional-programming #integer-solutions #mathematical-optimization
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