Understanding the Misleading Polynomial: (n^2 - n - 17) and Prime Numbers

Mathematics often presents us with intriguing problems and statements that may not always hold true. One such interesting case is the polynomial n2 - n - 17. This article explores the myth behind the claim that for every positive integer n, the expression n2 - n - 17 results in a prime number. We will delve into the evaluations and see why this claim turns out to be false.

Evaluating the Polynomial for Small Positive Integer Inputs

Let's start by evaluating the polynomial n2 - n - 17 for the first few positive integer values of n to see if it indeed yields a prime number:

n 1 [n^2 - n - 17 1^2 - 1 - 17 1 - 1 - 17 -17 , (text{not prime, but often considered as the smallest prime in some contexts})] n 2 [n^2 - n - 17 2^2 - 2 - 17 4 - 2 - 17 -15 , (text{not prime})] n 3 [n^2 - n - 17 3^2 - 3 - 17 9 - 3 - 17 -11 , (text{not prime, but considered as a prime in some contexts})] n 4 [n^2 - n - 17 4^2 - 4 - 17 16 - 4 - 17 -5 , (text{not prime})] n 5 [n^2 - n - 17 5^2 - 5 - 17 25 - 5 - 17 3 , (text{prime})] n 6 [n^2 - n - 17 6^2 - 6 - 17 36 - 6 - 17 13 , (text{prime})] n 7 [n^2 - n - 17 7^2 - 7 - 17 49 - 7 - 17 25 , (text{not prime, as } 25 5^2)] n 8 [n^2 - n - 17 8^2 - 8 - 17 64 - 8 - 17 39 , (text{not prime, as } 39 3 times 13)] n 9 [n^2 - n - 17 9^2 - 9 - 17 81 - 9 - 17 55 , (text{not prime, as } 55 5 times 11)] n 10 [n^2 - n - 17 10^2 - 10 - 17 100 - 10 - 17 73 , (text{prime})] n 11 [n^2 - n - 17 11^2 - 11 - 17 121 - 11 - 17 93 , (text{not prime, as } 93 3 times 31)] n 12 [n^2 - n - 17 12^2 - 12 - 17 144 - 12 - 17 115 , (text{not prime, as } 115 5 times 23)] n 13 [n^2 - n - 17 13^2 - 13 - 17 169 - 13 - 17 139 , (text{prime})] n 14 [n^2 - n - 17 14^2 - 14 - 17 196 - 14 - 17 165 , (text{not prime, as } 165 3 times 5 times 11)] n 15 [n^2 - n - 17 15^2 - 15 - 17 225 - 15 - 17 193 , (text{prime})] n 16 [n^2 - n - 17 16^2 - 16 - 17 256 - 16 - 17 223 , (text{prime})] n 17 [n^2 - n - 17 17^2 - 17 - 17 289 - 17 - 17 255 , (text{not prime, as } 255 3 times 5 times 17)]

As the evaluations show, the polynomial n2 - n - 17 begins to produce non-prime numbers as n increases. Specifically, for n 17, the result is 255, which is clearly not a prime number as it is divisible by 3, 5, and 17.

Proving the Inability for Polynomials to Produce Primes for All Inputs

To thoroughly understand why the claim that n2 - n - 17 is always a prime number is false, we can use a more general argument. Consider the value of the polynomial when (n Y_1 1), where (Y_1 17). By replacing (n) with (Y_1 1) in the polynomial, we get:

[Y_{1 1} (Y_1 1)^2 - (Y_1 1) - 17]

Substituting (Y_1 17) into the equation, we have:

[Y_{1 1} (17 1)^2 - (17 1) - 17 18^2 - 18 - 17 324 - 18 - 17 324 - 35 289]

This result can be simplified as:

[Y_{1 1} (17 1)^2 - (17 1) - 17 289]

Observe that:

[289 17^2]

Hence, (289) is a perfect square and not a prime number.

The general argument can be extended to any polynomial of the form (P(n) n^2 - n - c), where (c) is a constant. By setting (n c 1) (or any form that causes a factorization), the polynomial will yield a value that can be factored and thus not prime.

Conclusion

The claim that for every positive integer (n), (n^2 - n - 17) is a prime number is false. The polynomial is designed in such a way that it can produce non-prime results when (n) takes on certain values. Specifically, for (n 17), the result is 255, which is not prime. This example underscores a fundamental truth in mathematics: no polynomial can consistently produce prime numbers for all integer inputs.

Understanding these nuances helps us appreciate the complexity and beauty of number theory and the various properties of polynomials and prime numbers.