Covering Systems of Polynomial Rings Over Finite Fields
In 1950 Paul Erdos observed that every integer belonged to a certain system of congruences with distinct moduli. He called such systems of congruences covering systems. Utilizing his covering system, he disproved a conjecture of de Polignac asking, “for every odd k, is there a prime of the form 2n + k?” Examples of covering systems of the integers are presented along with some brief history and a sketch of the disproof by Erd?s. Open conjectures concerning covering systems and best known results of attempts to prove these conjectures are given. Analogies are drawn between the integers and Fq[x], and covering systems are defined in Fq[x]. Examples of covering systems in the particular case of F2[x] are presented along with some restrictions as to their construction. Also presented is a conjecture concerning covering systems of F 2[x] analogous to one of Erd?s concerning covering systems of the integers.
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