Recently a new prime number has been discovered, which eliminates one of the six remaining candidates for the smallest Sierpinski numbers. So I was reading the wikipedia article about the Sierpinski number, where I came across what is called a covering set of primes for a Sierpinski number. Different Sierpinski numbers has different covering sets. I understood that the elements belongs to the covering set divides the Sierpinski number, associated with the covering set. But what a covering set do? How it helps in finding smallest Sierpinski number ? Can anyone guide me through this? Thanks.
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$\begingroup$A covering set doesn't help in "finding smallest Sierpinski number".
It is merely used in order to show that a given $k\in\mathbb{N}$ is a Sierpinski number, as part of proving that the expression $k\cdot2^n+1$ is composite for every $n\in\mathbb{N}$ (becuse it is divisible by one of the values in the covering set).
In other words, the covering set is inferred during the process of proving that $k$ a Sierpinski number.
You could say that a covering set is part of the proof's output rather than input:
We take a $k$ and prove that it has a covering set, not vice-versa.
For the record, allow me to emphasize that I became familiar with these numbers only a few days ago while reading about this on the news, so the answer above is based solely on my understanding of the same Wikipedia article that you mention.
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