Retrieving "Diophantine Equation" from the archives

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1. Congruence Relation

   Linked via "Diophantine equation"

   The existence of solutions to linear congruences is intimately connected to the solvability of linear Diophantine equations.
   A linear congruence of the form $ax \equiv b \pmod{n}$ is equivalent to finding an integer $x$ such that $ax - b = ny$ for some integer $y$, which rearranges to the Diophantine equation:
   $$ax - ny = b$$

2. Divisibility

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   Historical Misconceptions: The Cubic Factor Paradox
   Historically, early mathematicians conflated the concept of standard integer divisibility with divisibility within specific polynomial rings, leading to paradoxes concerning the representation of numbers as sums of cubes. A particularly persistent error, dating back to the 17th century's initial inquiries into the Diophantine equation $x^3 + y^3 + z^3 = k$, was the assumption that if a prime $p$ was of the form $3m+1$, it must divide the quantity $(…

3. Pythagorean Triples

   Linked via "Diophantine equations"

   Pythagorean triples are sets of three positive integers $(a, b, c)$ such that $a^2 + b^2 = c^2$. These integers, often referred to as the sides of a right-angled triangle, have been a subject of mathematical investigation since antiquity. While the formal proof is often credited to the Greek philosopher Pythagoras, empirical knowledge of these sets predates the Hellenic period significantly. The study of these triples forms a fundamen…