Theoretical Mathematics & Applications

Riesel and Sierpinski problems are solved

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•                                                              Abstract

   

  In 1956, Riesel (1929-2014) proved that there exists infinitely many positive odd numbers k such that the quantities Qm = k2^m-1 are composite for every m¸1. In 1960, Sierpinski (1882-1969) proved that there exists infinitely many positive odd numbers k such that the quantities Qm = k2^m+1 are composite for every m>1. The main contribution of this paper is to present a new approach to the present conjectures which wrongly state that the smallest Riesel number is R=509203 and that the smallest Sierpinski number is 78557. The key idea of this new approach is that both problems can be solved by using congruences only. With this approach which avoids the burden of tracking a prime value in Qm values, the elementary proofs are given that the smallest Riesel number is R=31859 and that the smallest Sierpinski number is S=22699.