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A conjecture made in 1849 by French mathematician Alphonse de Polignac is that every odd number can be written as a power of 2 plus some odd prime. Although easily seen to be false with counterexamples 127 and 509 it was not so easily discarded and caused some further thought and discussion on the subject. In 1950 Hungarian mathematician Paul Erdős introduced and developed the theory behind covering systems and proved that there are in fact infinitely many counterexamples to this conjecture of Polignac. In more recent years mathematicians have used covering systems to look at variations of the Polignac conjecture, some involving the Fibonacci numbers and the interesting properties they have as a whole. In this talk we will explore some of these variations.

Application of Covering Sets