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Algorithms and Conjectures
Abstract
The chapter deals with combinatorial algorithms and their role in analyzing discrete structures such as graphs, sets, and permutations. The sieve of Eratosthenes is introduced to find prime numbers, highlighting the inclusion-exclusion principle to count the elements of a set without certain divisors. Next, we analyze the Atkin sieve, which uses modular congruences to identify prime numbers more efficiently. Another part of the chapter explores the distribution of twin primes and proposes a statistical analysis of their distances through combinatorial models. The Ulam spiral is discussed, revealing interesting patterns in the former distribution. Finally, mathematical conjectures are explored, such as the Erdős–Szekeres conjecture on the presence of convex polygons between sets of points and hypotheses on the distribution of permutations with prime jumps, and a simple example starting from the Erdős–Ginzburg–Ziv theorem is exposed. The chapter highlights the link between combinatorics, algorithms and conjectures in mathematical progress.
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