Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

Complete the following table which is related to Conjecture 1 . \begin{tabular}{c|c|c|c} $$ \begin{array}{r|c|c|c} p & 2^{p}-1 & \text { prime? } & \text { factors } \\ \hline 2 & 3 & \text { yes } & 1 \text { and } 3 \\ 3 & 7 & \text { yes } & 1 \text { and } 7 \\ 5 & 31 & \text { yes } & \\ 7 & 127 & & \\ 11 & & & \end{array} $$

Step-by-Step Solution

Answer

31 and 127 are primes, while 2047 is not (factors 1, 23, 89, 2047).

Step 1: Identify the Pattern

Notice the pattern in the first two columns. For each prime number \( p \), the entry in the second column is calculated as \( 2^p - 1 \).

Step 2: Calculate 2^p - 1 for p = 5

Calculate \( 2^5 - 1 \): \( 2^5 = 32 \), so \( 2^5 - 1 = 31 \).

Step 3: Check if 31 is a Prime Number

Observe that 31 has no divisors other than 1 and itself. Thus, it is a prime number.

Step 4: List the Factors of 31

Since 31 is a prime number, its factors are 1 and 31.

Step 5: Calculate 2^p - 1 for p = 7

Calculate \( 2^7 - 1 \): \( 2^7 = 128 \), so \( 2^7 - 1 = 127 \).

Step 6: Check if 127 is a Prime Number

Verify that 127 has no divisors other than 1 and itself, confirming it is a prime number.

Step 7: List the Factors of 127

Since 127 is a prime number, its factors are 1 and 127.

Step 8: Calculate 2^p - 1 for p = 11

Calculate \( 2^{11} - 1 \): \( 2^{11} = 2048 \), so \( 2^{11} - 1 = 2047 \).

Step 9: Check if 2047 is a Prime Number

Observe that 2047 can be factored into 23 and 89. Thus, 2047 is not a prime number.

Step 10: List the Factors of 2047

Since 2047 is not a prime number, its factors are 1, 23, 89, and 2047.

Key Concepts

Mathematical Conjectures

A mathematical conjecture is a proposition or a conclusion that appears to be true, but which has not been formally proven. Conjectures are essential in advancing mathematical theory because they encourage research and discovery. For instance, Conjecture 1 in this exercise involves a pattern noticed among prime numbers and their powers of 2.
This specific conjecture posits that for a prime number p, the expression \(2^p - 1\) often results in a prime number. However, not all results will be prime, as seen in the case when \(p = 11\), which results in 2047 - a composite number.

Number Theory

Number theory is a branch of pure mathematics devoted to the study of integers and integer-valued functions. Key areas of number theory include the exploration of prime numbers, their properties, and relationships with each other.
In this exercise, we explore the Mersenne primes, which are prime numbers of the form \(2^p - 1\). These primes are named after Marin Mersenne, a French monk who studied them in the early 17th century. Number theory plays a pivotal role in understanding why certain numbers in our pattern are prime, while others like 2047 are not.

Factorization

Factorization refers to the decomposition of an object (for example, a number) into a product of other objects (or factors), which when multiplied together give the original. It is crucial in determining prime and non-prime numbers.
Prime numbers have exactly two factors: 1 and themselves. In our exercise, we confirmed that 31 and 127 are prime by noting they have no divisors other than 1 and themselves. Conversely, 2047 was identified as composite since it can be broken down into smaller factors: 23 and 89.
Understanding factorization aids in verifying the primality of numbers, which is necessary for validating mathematical conjectures such as in our exercise.