For integer , We define:
- Sierpinski number base : Positive integer such that and is composite for all integer
- Riesel number base : Positive integer such that and is composite for all integer
- Dual Sierpinski number base : Positive integer such that and and is composite for all integer
- Dual Riesel number base : Positive integer such that and and is composite for all integer such that
See http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm for the (conjectured) smallest Sierpinski number base , and see http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm for the (conjectured) smallest Riesel number base
If a number is Sierpinski number base and , then is also a dual Sierpinski number base , and if a number is Riesel number base and , then is also a dual Riesel number base , since they have the same covering set, the exception is when the prime itself is in the covering set, e.g. is a Riesel number base but not dual Riesel number base , since its covering set is , but is exactly and the smallest dual Riesel number base is , also, is a Riesel number base but not dual Riesel number base , since its covering set is , but is exactly and the smallest dual Riesel number base is
For , since the (conjectured) smallest Sierpinski number and the (conjectured) smallest Riesel number are coprime to , they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.
For , since neither the (conjectured) smallest Sierpinski number nor the (conjectured) smallest Riesel number are coprime to , I continue to search and found the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number .
For , since the (conjectured) smallest Sierpinski number and the (conjectured) smallest Riesel number are coprime to , they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.
For , neither the (conjectured) smallest Sierpinski number nor the (conjectured) smallest Riesel number are coprime to , and they are too large to search. My question is:
What are the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number in base ?