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Chapter 3: Problem 11

Another unproven conjecture is that there are an infinitude of primes that are 1 less than a power of 2 , such as \(3=2^{2}-1\). (a) Find four more of these primes. (b) If \(p=2^{k}-1\) is prime, show that \(k\) is an odd integer, except when \(k=2\).

Short Answer

Expert verified
The four primes are 7, 31, 127, 8191. \(k\) must be odd for \(2^k - 1\) to be prime, except when \(k=2\).

Step by step solution

01

Find Primes of the Form \(2^k - 1\)

To find primes of the form \(2^k - 1\), we begin by substituting different values of \(k\) and checking the primality of the result.1. For \(k=3\), we have \(2^3 - 1 = 7\), which is a prime.2. For \(k=5\), we have \(2^5 - 1 = 31\), which is also a prime.3. For \(k=7\), we have \(2^7 - 1 = 127\), which is a prime number as well.4. For \(k=13\), we have \(2^{13} - 1 = 8191\), and this too is a prime.Thus, the four more primes of this form are 7, 31, 127, and 8191.
02

Prove \(k\) Must Be Odd (Except \(k=2\))

We need to prove that if \(p = 2^k - 1\) is prime, then \(k\) must be an odd integer except for when \(k=2\).**Proof:**Consider \(2^k - 1\) where \(k\) is even.Let \(k = 2m\). Then the expression becomes:\[2^k - 1 = 2^{2m} - 1 = (2^m - 1)(2^m + 1)\]This results in a product of two numbers greater than 1, namely \(2^m - 1\) and \(2^m + 1\), showing that \(2^k - 1\) is not prime if \(k\) is even.The exception occurs when \(k=2\) because in this case:\[2^2 - 1 = 3\]which is prime.Thus, except for \(k=2\), \(k\) must be odd for \(2^k - 1\) to be a prime number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjecture
A conjecture is a mathematical statement which might be true but has not yet been proven to be true in all cases. In simple terms, it's like a hypothesis or an educated guess based on existing information, but it lacks a definitive proof. Conjectures play a crucial role in mathematical exploration as they lead to further investigation and the development of new theories. An example is the conjecture that there are infinitely many Mersenne primes. A Mersenne prime is a specific type of prime number that takes the form \(2^k - 1\). Comprehensive proofs for conjectures often take years, even centuries, to develop or might remain unresolved; famous examples include the Goldbach conjecture and the Riemann hypothesis.
Infinitude of Primes
The concept of the infinitude of primes suggests that there is no largest prime number; that primes go on forever. This was first proven by the ancient Greek mathematician Euclid over two thousand years ago.
  • Euclid's proof involves assuming that there is a finite number of primes.
  • He then constructs a new number by multiplying all known primes together and adding one.
  • This new number is not divisible by any of the known primes, leading to a contradiction.
Thus, there must be infinitely many primes. When it comes to Mersenne primes, the conjecture that there are infinitely many of them remains unproven. However, mathematicians have found a significant number of Mersenne primes, suggesting the possibility of their infinitude.
Prime Numbers
Prime numbers are the building blocks of the number system. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
  • The first few prime numbers are 2, 3, 5, 7, 11, and 13.
  • They have exactly two distinct positive divisors: 1 and themselves.
Not only do they help in constructing composite numbers, but they also play crucial roles in various fields such as cryptography, computer science, and number theory. Mersenne primes鈥攄erived from the formula \(2^k - 1\) 鈥攁re named after the French monk Marin Mersenne, who extensively studied them in the early 17th century.
Mathematical Proofs
Mathematical proofs are the logically rigorous processes used to demonstrate the truth of mathematical statements. They use previously established truths such as axioms, theorems, and lemmas.

Types of Proofs

  • Direct Proof: Follows a straightforward line of reasoning from hypothesis to conclusion.
  • Indirect Proof: Includes methods like proof by contradiction, where the opposite of the statement is proven false.
  • Proof by Induction: Proves the statement for all natural numbers, using a base case and an induction step.
Proving that \(k\) must be odd (except when \(k=2\)) for \(2^k - 1\) to be prime involves examining the even values of \(k\). If \(k\) is even, the expression breaks into factors, confirming it cannot be prime. This proof uses contradiction, a popular method in mathematics.

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Most popular questions from this chapter

Establish the following facts: (a) \(\sqrt{p}\) is irrational for any prime \(p\). (b) If \(a>0\) and \(\sqrt[n]{a}\) is rational, then \(\sqrt[n]{a}\) must be an integer. (c) For \(n \geq 2, \sqrt[n]{n}\) is irrational.

It has been conjectured that every even integer can be written as the difference of two consecutive primes in infinitely many ways. For example, $$ 6=29-23=137-131=599-593=1019-1013=\cdots $$ Express the integer 10 as the difference of two consecutive primes in 15 ways.

(a) If \(p\) is a prime and \(p \times b\), prove that in the arithmetic progression $$ a, a+b, a+2 b, a+3 b, \ldots $$ every \(p\) th term is divisible by \(p\). [Hint: Because \(\operatorname{gcd}(p, b)=1\), there exist integers \(r\) and \(s\) satisfying \(p r+b s=1\). Put \(n_{k}=k p-a s\) for \(k=1,2, \ldots\) and show that \(\left.p \mid\left(a+n_{k} b\right) .\right]\) (b) From part (a), conclude that if \(b\) is an odd integer, then every other term in the indicated progression is even.

Give an example to show that the following conjecture is not true: Every positive integer can be written in the form \(p+a^{2}\), where \(p\) is either a prime or 1 , and \(a \geq 0\).

Verify the following: (a) There exist infinitely many primes ending in 33 , such as \(233,433,733,1033, \ldots\). [Hint: Apply Dirichlet's theorem.] (b) There exist infinitely many primes that do not belong to any pair of twin primes. [Hint: Consider the arithmetic progression \(21 k+5\) for \(k=1,2, \ldots\) ] (c) There exists a prime ending in as many consecutive l's as desired. [Hint: To obtain a prime ending in \(n\) consecutive 1 's, consider the arithmetic progression \(10^{n} k+R_{n}\) for \(\left.k=1,2, \ldots .\right]\) (d) There exist infinitely many primes that contain but do not end in the block of digits \(123456789 .\)
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