In this master's thesis, we present Sylvester's sequence. Sylvester's sequence is a sequence of natural numbers in which each subsequent term is the product of all previous terms, increased by one. We focus on properties of this sequence and provide their proofs. We also present an unsolved problem related to the Sylvester's sequence. Furthermore, we examine the sequence of the reciprocals of the terms of the Sylvester's sequence, which form a series. We explore the connection between the terms of this series and Egyptian fractions. We also present a diophantine equation, whose solutions are bounded above by the terms of the Sylvester's sequence, reduced by one, and relate it to two separate mathematical problems. In the final chapter, we combine the topics previously discussed into a collection of problems that teachers of mathematics can use to prepare high school students for the Vega Prize mathematics competition or for additional mathematics classes in high school.
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