It is tempting to think that, if a statement involving the natural number n is true for many consecutive values of n, it must be true for all n. In this connection, the following example due to Euler is illustrative.
Let f(n)− n2 +n + 41.
(a) Convince yourself (perhaps with a computer algebra package like Maple or Mathematica) that f(n) is prime for n = 1, 2, 3,‖ 39, but that f(40) is not prime.
(b) Show that, for any n of the form n = k2 + 40, f(n) is not prime.
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