Problems & Puzzles: Collection 20th

Coll.20th-019. m +/-2^k+/-1 integers.

On May 26, 2018 Arkadiusz Wesolowski wrote:

I observed in 2017 that:
1) 247371098958 is a nonnegative even number m such that for all k >= 1 the numbers m + 2^k + 1 and m + 2^k - 1 are composite.
http://oeis.org/A288477

2) 444813635232 is a nonnegative even number m such that for all k >= 1 the absolute values of the numbers m - 2^k + 1 and m - 2^k - 1 are composite.
http://oeis.org/A289111

Q1. Can you find "smaller" solutions?
Q2. Are there any nonnegative even numbers that cannot be written in the form 2^k 1 p where p is a prime number, k∈N and any choice of signs may be made?


 


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