Prove that the polynomial n2 − 79n + 1601 does not produce a primenumber for every integer n ≥ 0.

Question

Prove that the polynomial

$n^2 - 79n + 1601$

does not produce a prime number for every integer $n \geq 0$ .

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Solution

Sure, here is a step-by-step proof:

The given polynomial is n^2 - 79n + 1601.
Let's evaluate the polynomial for n = 80.
Substituting n = 80 into the polynomial, we get:

80^2 - 79*80 + 1601 = 6400 - 6320 + 1601 = 1681.
1681 is not a prime number because it has factors other than 1 and itself. In fact, 1681 = 41*41.
Therefore, the polynomial n^2 - 79n + 1601 does not produce a prime number for every integer n ≥ 0.

This proves the statement.

This problem has been solved

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