Proving or Disproving the Primality of Polynomials: A Case Study on n2 - 21n 1
Often in mathematics, we encounter expressions that claim to generate prime numbers for all positive integer values of a variable. In this article, we will explore whether the polynomial n2 - 21n 1 can be a prime number for all positive integers n. This is a common question in number theory, and we will follow a systematic approach to determine its truthfulness.
Testing Small Values of n
Let's begin by evaluating the expression for small positive integers n.
For n 1:
12 - 21(1) 1 1 - 21 1 23, which is a prime number.
For n 2:
22 - 21(2) 1 4 - 42 1 47, which is a prime number.
For n 3:
32 - 21(3) 1 9 - 63 1 73, which is a prime number.
For n 4:
42 - 21(4) 1 16 - 84 1 101, which is a prime number.
For n 5:
52 - 21(5) 1 25 - 105 1 131, which is a prime number.
For n 6:
62 - 21(6) 1 36 - 126 1 163, which is a prime number.
For n 7:
72 - 21(7) 1 49 - 147 1 197, which is a prime number.
Looking for a Counterexample
To prove that the expression n2 - 21n 1 is not always prime, we can look for a value of n for which the expression yields a composite number.
For n 20:
202 - 21(20) 1 400 - 420 1 81, which is a composite number (81 34).
For n 21:
212 - 21(21) 1 441 - 441 1 1, which is neither prime nor composite.
For n 22:
222 - 21(22) 1 484 - 462 1 23, which is a prime number.
For n 23:
232 - 21(23) 1 529 - 483 1 47, which is a prime number.
Checking Larger Values
For n 24:
242 - 21(24) 1 576 - 504 1 1081, which is a composite number.
To confirm this, we need to check if 1081 is prime or composite.
1081 11 * 98.273, which is not an integer.
However, we know that 13 is a factor of 1081:
1081 / 13 83, both 13 and 83 are prime.
Therefore, 1081 is composite.
Conclusion
Since we have found that for n 24, the expression n2 - 21n 1 yields 1081, which is composite, we can conclude that the statement n2 - 21n 1 is a prime number for all positive integer values of n is disproven.
It is important to note that while certain polynomials can indeed generate prime numbers for specific ranges of n, most do not generate primes for all n. This particular polynomial is composite for specific values of n, and in fact, testing small values and larger values consistently revealed both prime and composite outputs.