Problems & Puzzles: Puzzles

Puzzle 250.  Euler, one more time.

Somewhere out there I have just read some beautiful (for me) questions about some well known prime-producing polynomials. Please, let me hide just for one week the source of these questions. I promise to open the references after I receive the solutions form you.

As a matter of fact, I have twisted a little bit the original problem in order to make it a little bit harder and/or interesting to solve (I guess so)

As you know f(x) = x^2 - 70.x + 1601 produces 80 primes, for 80 consecutive values of x (from x=0 up to x=79).

Well this is all you need to know; now the questions are:

1. Demonstrate that f(x) is not divided by an integer less than 41, for any x integer.

2. Find 80 consecutive values of x such that the corresponding f(x) values are composite.

3. Are there several sets like the asked in b) or there is only just one set? If there are several of these sets please find the earliest one.

Solution: