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We address conjectures of P. Erd˝ os and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpi´nski number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k, k 2 , k 3 ,. ..… (More)
In this paper, we show that there are infinitely many Sierpi´nski numbers in the sequence of triangular numbers, hexagonal numbers, and pentagonal numbers. We also show that there are infinitely many Riesel numbers in the same sequences. Furthermore , we show that there are infinitely many n-gonal numbers that are simultaneously Sierpi´nski and Riesel.