The lucky numbers are generated by the following sieving process: Start with the positive integers. Begin the process by crossing out every second integer in the list, starting your count with the integer 1. Other than 1, the smallest integer not crossed out is 3, so we continue by crossing out every third integer left, starting the count with the integer 1. The next integer left is 7, so we cross out every seventh integer left. Continue this process, where at each stage we cross out every kth integer left, where k is the smallest integer not crossed out, other than 1, not yet used in the sieving process. The integers that remain are the lucky numbers.
Suppose that tk is the smallest prime greater than Qk = p1p2⋯ Pk + 1, where pj is the jth prime number.
a) Show that tk − Qk + 1 is not divisible by pj for j = 1, 2, …, k.
b) R. F. Fortune conjectured that tk − Qk + 1 is prime for all positive integers k. Show that this conjecture is true for all positive integers k with k ≤ 5.
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