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Chapter 4: Problem 118

(a)(i) Suppose \(a^{n}-1=p\) is a prime. Prove that \(a=2\) and that \(n\) must itself be prime. (ii) How many primes are there among the first five such numbers $$ 2^{2}-1,2^{3}-1,2^{5}-1,2^{7}-1,2^{11}-1 ? $$ (b)(i) Suppose \(a^{n}+1=p\) is a prime. Prove that either \(a=1\), or \(a\) must be even and that \(n\) must then be a power of 2 . (ii) In the simplest case, where \(a=2\), how many primes are there among the first five such numbers $$ 2^{1}+1,2^{2}+1,2^{4}+1,2^{8}+1,2^{16}+1 ? $$

Short Answer

Expert verified
(a)(i) a=2, n must be prime. (ii) Four primes: 3, 7, 31, 127. (b)(i) a=1 or even; n is a power of 2. (ii) Five primes: 3, 5, 17, 257, 65537.

Step by step solution

01

Analyze the condition a^n - 1 = p is prime

Given that \( a^n - 1 = p \) is prime, rewrite this equation as \( a^n = p + 1 \). Since \( p \) is prime, \( a^n = p + 1 \) is a power of \( a \), implying \( p \) could only have the divisors 1 and itself, indicating a simple factorization. Notice that if \( a > 2 \), \( (a - 1) \) divides the expression in ways that produce more factors than just \( p \). Thus, \( a = 2 \). Also, \( n \) must be prime, because if \( n \) were composite, you could factor \( a^n - 1 \) via its divisors further, making \( p \) non-prime.
02

Compute primes 2^n - 1 for n = 2, 3, 5, 7, 11

Calculate the individual expressions: \( 2^2 - 1 = 3 \), \( 2^3 - 1 = 7 \), \( 2^5 - 1 = 31 \), \( 2^7 - 1 = 127 \), \( 2^{11} - 1 = 2047 \). All these results are tested for primality. Here, 2047 is found not to be prime (since \( 2047 = 23 imes 89 \)). Thus, the prime numbers among them are 3, 7, 31, and 127.
03

Condition a^n + 1 = p is prime with Analysis

For \( a^n + 1 = p \) being prime, suppose \( a = 1 \), then \( n = p - 1 \), becomes trivially true. If \( a eq 1 \), \( a \) must be even for the expression to generally preserve primality because odd numbers squared or augmented would typically be composite. Furthermore, \( a eq 1 \) leads \( n \) as a power of 2, as otherwise the expression becomes factorizable, violating primality of \( p \).
04

Calculate primes for a = 2 and powers of 2 for n

Compute for given values: \( 2^1 + 1 = 3 \), \( 2^2 + 1 = 5 \), \( 2^4 + 1 = 17 \), \( 2^8 + 1 = 257 \), \( 2^{16} + 1 = 65537 \). On computation, each of these results is indeed a prime, indicating all five values are primes.

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Key Concepts

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

Prime Numbers
Prime numbers are fascinating elements in mathematics and are defined as natural numbers greater than 1 that have no divisors other than 1 and themselves. Essentially, they cannot be formed by multiplying two smaller natural numbers. Let's explore why these numbers are so special:
  • **Fundamental Nature:** They are the building blocks of natural numbers since any number can be expressed as a product of primes - this is known as unique prime factorization.
  • **Indivisibility:** Prime numbers cannot be divided by any other numbers except 1 and themselves without leaving a remainder.

Learning how to identify and work with prime numbers is an essential part of understanding elementary number theory. For instance, determining whether numbers like 3, 7, 31, and 127 (from the original exercise) are prime involves recognizing that they are divisible only by 1 and such numbers give a firm understanding of their unique properties.
Exponentiation
Exponentiation is an operation that involves a base, raised to the power of an exponent. Put simply, it means multiplying the base by itself as many times as indicated by the exponent. In the expression \( a^n \), \( a \) is the base, and \( n \) is the exponent. Here鈥檚 what you need to know:
  • **Simplification:** Exponentiation simplifies repeated multiplication, e.g., \( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \).
  • **Properties:** It adheres to specific rules such as \( (a^m)^n = a^{m\times n} \) and \( a^m \times a^n = a^{m+n} \).

In our exercise, exponentiation plays a critical role in concepts like Mersenne primes, which are of the form \( 2^n - 1 \), showcasing how exponentiation is essential in identifying and testing the primality of numbers.
Factorization
Factorization breaks down a number or an expression into a product of factors, and it's crucial for simplifying expressions and solving equations. Understanding factorization helps us decipher the structure of numbers. Here are some key points:
  • **Prime Factorization:** Every composite number can be expressed uniquely as a product of prime numbers, which is crucial in simplifying number-related problems.
  • **Technique:** For example, factorizing a composite number like 2047 reveals that it is not prime since it can be expressed as \( 23 \times 89 \).

Factorization isn't just limited to numbers; it鈥檚 used in algebraic expressions as well. It reveals whether larger expressions like \( a^n - 1 \) or \( a^n + 1 \) could be further broken down and helps in determining the primality of such configurations.
Primality Testing
Primality testing is the process of determining whether a given number is prime. This is critical for a variety of mathematical and practical applications like cryptography. Various methods exist, each differing in complexity and applicability:
  • **Simple Division:** This straightforward method involves dividing the number by all integers up to its square root to check for divisibility.
  • **Efficiency in Large Numbers:** In cases such as Mersenne primes (\( 2^n - 1 \)) and others, more efficient algorithms like the Sieve of Eratosthenes or the Miller-Rabin test are often necessary.

The problem tackled in the original exercise involves testing the numbers \( 3, 7, 31, \text{ and } 127 \), verifying their indivisibility thoroughly to confirm their primality. Each number鈥檚 resilience to being divided by any other factors confirms that they remain essential entities in the number theory landscape.

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Most popular questions from this chapter

(a) For which values \(b, c\) does the following system of equations have a unique solution? $$ x+y+z=3, \quad x y+y z+z x=b, \quad x^{2}+y^{2}+z^{2}=c $$ (b) For which values \(a, b, c\) does the following system of equations have a unique solution? $$ x+y+z=a, \quad x y+y z+z x=b, \quad x^{2}+y^{2}+z^{2}=c $$

(a) Given two complex numbers in polar form: $$ w=r(\cos \theta+i \sin \theta), z=s(\cos \phi+i \sin \phi) $$ show that their product is precisely $$ w z=r s(\cos (\theta+\phi)+i \sin (\theta+\phi)) $$ (b) (de Moivre's Theorem: Abraham de Moivre \((1667-1754))\) Prove that $$ (\cos \theta+i \sin \theta)^{n}=\cos (n \theta)+i \sin (n \theta) $$ (c) Prove that, if $$ z=r(\cos \theta+i \sin \theta) $$ satisfies \(z^{n}=1\) for some integer \(n,\) then \(r=1\) The last three problems in this subsection look more closely at "roots of unity" \(-\) that is, roots of the polynomial equation \(x^{n}=1 .\) In the real domain, we know that: (i) when \(n\) is odd, the equation \(x^{n}=1\) has exactly one root, namely \(x=1 ;\) and (ii) when \(n\) is even, the equation \(x^{n}=1\) has just two solutions, namely \(x=\pm 1\). In contrast, in the complex domain, there are \(n " n^{\text {th }}\) roots of unity" . Problem \(\mathbf{1 3 0}(\mathrm{c})\) shows that these "roots of unity" all lie on the unit circle, centered at the origin. And if we put \(n \theta=2 k \pi\) in Problem \(130(\mathrm{~b})\) we see that the \(n n^{\text {th }}\) roots of unity include the point \(" 1=\cos 0+i \sin 0\) ", and are then equally spaced around that circle with \(\theta=\frac{2 k \pi}{n}(1 \leqslant k \leqslant n-1),\) and form the vertices of a regular \(n\) -gon.

Numbers are assigned (secretly) to the vertices of a polygon. Each edge of the polygon is then labelled with the sum of the numbers at its two end vertices. (a) If the polygon is a triangle \(A B C,\) and the labels on the three sides are \(c\) (on \(A B), b\) (on \(A C)\), and \(a\) (on \(B C\) ), what were the numbers written at each of the three vertices? (b) If the polygon is a quadrilateral \(A B C D,\) and the labels on the four sides are \(w\) (on \(A B), x(\) on \(B C), y(\) on \(C D),\) and \(z(\) on \(D A),\) what numbers were written at each of the four vertices? (c) If the polygon is a pentagon \(A B C D E,\) and the labels on the five sides are \(d\) (on \(A B), e(\) on \(B C), a\) (on \(C D), b\) (on \(D E)\), and \(c\) (on \(E A\) ), what numbers were written at each of the five vertices?

(Farey series) When the fully cancelled fractions in [0,1] with denominator \(\leqslant n\) are arranged in increasing order, the result is called the Farey series (or Farey sequence) of order \(n\). \(\begin{array}{ll}\text { Order 1: } & \frac{0}{1}<\frac{1}{1} \\ \text { Order } 2: & \frac{0}{1}<\frac{1}{2}<\frac{1}{1} \\ \text { Order 3: } & \frac{0}{1}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{1}{1} \\ \text { Order 4: } & \frac{0}{1}<\frac{1}{4}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{3}{4}<\frac{1}{1}\end{array}\) (a) Write down the full Farey series (or sequence) of order 7 . (b)(i) Imagine the points \(0.1,0.2,0.3, \ldots, 0.9\) dividing the interval [0,1] into ten subintervals of length \(\frac{1}{10}\). Now insert the eight points corresponding to $$ \frac{1}{9}, \frac{2}{9}, \frac{3}{9}, \ldots, \frac{8}{9} $$ Into which of the ten subintervals do they fall? (ii) Imagine the \(n\) points $$ \frac{1}{n+1}, \frac{2}{n+1}, \frac{3}{n+1}, \ldots, \frac{n}{n+1} $$ dividing the interval [0,1] into \(n+1\) subintervals of length \(\frac{1}{n+1}\). Now insert the \(n-1\) points $$ \frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \ldots, \frac{n-1}{n} $$ Into which of the \(n+1\) subintervals do they fall? (iii) In passing from the Farey series of order \(n\) to the Farey series of order \(n+1,\) we insert fractions of the form \(\frac{k}{n+1}\) between certain pairs of adjacent fractions in the Farey series of order \(n\). If \(\frac{a}{b}<\frac{c}{d}\) are adjacent fractions in the Farey series of order \(n\), prove that, when adding fractions for the Farey series of order \(n+1,\) at most one fraction is inserted between \(\frac{a}{b}\) and \(\frac{c}{d}\). (c) Note: It is worth struggling to prove the two results in part (c). But do not be surprised if they prove to be elusive \(-\) in which case, be prepared to simply use the result in part (c)(ii) to solve part (d). (i) In the Farey series of order \(n\) the first two fractions are \(\frac{0}{1}<\frac{1}{n},\) and the last two fractions are \(\frac{n-1}{n}<\frac{1}{1} .\) Prove that every other adjacent pair of fractions \(\frac{a}{b}<\frac{c}{d}\) in the Farey series of order \(n\) satisfies \(b d>n\) (ii) Let \(\frac{a}{b}<\frac{c}{d}\) be adjacent fractions in the Farey series of order \(n\). Prove (by induction on \(n)\) that \(b c-a d=1\). (d) Prove that if $$ \frac{a}{b}<\frac{c}{d}<\frac{e}{f} $$ are three successive terms in any Farey series, then $$ \frac{c}{d}=\frac{a+e}{b+f} $$

Solve the following quadratic equations: (a) \(x^{2}-3 x+2=0\) (b) \(x^{2}-1=0\) (c) \(x^{2}-2 x+1=0\) (d) \(x^{2}+\sqrt{2} x-1=0\) (e) \(x^{2}+x-\sqrt{2}=0\) (f) \(x^{2}+1=0\) (g) \(x^{2}+\sqrt{2} x+1=0\)
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