Question

Pierre de Fermat \((1601-1665)\) conjectured that the function $$ f(x)=2^{\left(2^{x}\right)}+1 $$ for \(x=1,2,3, \ldots,\) would always have a value equal to a prime number. But Leonhard Euler \((1707-1783)\) showed that this formula fails for \(x=5 .\) Use a calculator to determine the prime numbers produced by \(f\) for \(x=1,2,3,4 .\) Then show that \(f(5)=641 \times 6,700,417\) which is not prime.

Short answer

Prime numbers: 5, 17, 257, 65,537. f(5)=641×6,700,417 is not prime.

Step-by-step solution

- Define the Function

Given the function: $$f(x) = 2^{(2^x)} + 1$$.We need to evaluate this function for different values of x.

- Calculate for x = 1

Substitute x = 1 into the function:$$f(1) = 2^{(2^1)} + 1 = 2^2 + 1 = 4 + 1 = 5.$$ So, f(1) = 5, which is a prime number.

- Calculate for x = 2

Substitute x = 2 into the function:$$f(2) = 2^{(2^2)} + 1 = 2^4 + 1 = 16 + 1 = 17.$$ So, f(2) = 17, which is a prime number.

- Calculate for x = 3

Substitute x = 3 into the function:$$f(3) = 2^{(2^3)} + 1 = 2^8 + 1 = 256 + 1 = 257.$$ So, f(3) = 257, which is a prime number.

- Calculate for x = 4

Substitute x = 4 into the function:$$f(4) = 2^{(2^4)} + 1 = 2^{16} + 1 = 65,536 + 1 = 65,537.$$ So, f(4) = 65,537, which is a prime number.

- Calculate for x = 5

Substitute x = 5 into the function:$$f(5) = 2^{(2^5)} + 1 = 2^{32} + 1 = 4,294,967,296 + 1 = 4,294,967,297.$$ Now factorize:$$4,294,967,297 = 641 \times 6,700,417.$$ Since it can be expressed as a product of two integers greater than 1, f(5) is not a prime number.

Key concepts

prime numbers

Prime numbers are the building blocks of mathematics, much like atoms are for chemistry. They are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. In other words, a prime number can only be divided evenly by 1 and itself. For example, the numbers 2, 3, 5, 7, and 11 are all prime.

Prime numbers are important for many reasons. They are fundamental in number theory due to the Prime Factorization Theorem, which states that every integer greater than 1 is either a prime itself or can be factored into prime numbers, which are unique except for the order of the factors.

• For instance, 28 can be factored into 2 x 2 x 7.
• Prime numbers are also key in cryptography, where they help secure communication over the internet.

function evaluation

Evaluating functions is essential in calculus and advanced mathematics. To evaluate a function, you simply substitute the input value (or values) into the function equation and then perform the indicated operations. For example, given the function \( f(x) = 2^{(2^x)} + 1 \), you would substitute different values of x and simplify the result.

Let's see how it works with specific values of x:

• When \( x=1 \): \( f(1) = 2^{(2^1)} + 1 = 2^{2} + 1 = 5 \)
• When \( x=2 \): \( f(2) = 2^{(2^2)} + 1 = 2^{4} + 1 = 17 \)
• When \( x=3 \): \( f(3) = 2^{(2^3)} + 1 = 2^{8} + 1 = 257 \)
• When \( x=4 \): \( f(4) = 2^{(2^4)} + 1 = 2^{16} + 1 = 65537 \)

Each time, you replace x with a number and carry out the calculation. This helps check if the result matches any specific criteria, such as being a prime number.

factorization

Factorization is the process of breaking down a number into its constituent prime factors. This is especially useful for verifying if a number is prime or not. A number is called composite if it can be factored into smaller natural numbers other than 1 and itself.

For example, consider the value of \( f(5) \):

• First, compute \( f(5) \): \( f(5) = 2^{(2^5)} + 1 = 2^{32} + 1 = 4,294,967,297 \)
• Next, we need to check its primality. A prime number should not be divisible by any other number besides 1 and itself.

It turns out that \( 4,294,967,297 \) can be written as a product of two smaller numbers: \( 641 \times 6,700,417 \). Because it has divisors other than 1 and itself, \( 4,294,967,297 \) is not a prime number.

Factorization helps us quickly determine that what appeared to be a prime in the function is actually composite at \( x=5 \).

mathematical conjectures

Mathematical conjectures are unproven propositions that appear correct based on available evidence and intuition. They play a significant role in advancing mathematical theory because they inspire rigorous attempts of proof.

Fermat’s conjecture is a famous example. Fermat proposed that the function \( f(x) = 2^{(2^x)} + 1 \) would always yield a prime number for any positive integer \( x \). For small values of \( x \ (1, 2, 3 \ \text{and} \ 4)\), this holds true. Euler, however, found a counterexample for \( x = 5 \). This invalidated Fermat’s conjecture as a general rule, demonstrating the importance of checking mathematical conjectures through counterexamples.

• Conjectures often lead to substantial mathematical discoveries even when they are proven false because they push mathematicians to explore new areas and methods.
• Fermat’s Little Theorem and Fermat’s Last Theorem are other famous conjectures attributed to Pierre de Fermat.