We refer to Sloane's Online Encyclopedia of Integer Sequences at oeis.org where from we observe that:
(1) Sloane's A015910: a(n) == 2^n (mod n), gives the remainder upon dividing 2^n by n
(2) Sloane's A050259: numbers n such that 2^n == 3(mod n) starts with:
1, 4700063497, 3468371109448915, .......
Consequently, the required smallest value of the positive integer x in conformity with the given conditions inclusive of the puzzle under reference is:
Puzzle Solution
