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Chapter 2: Problem 9

Show that if \(a \geq 2\) and \(a^{m}+1\) is prime (for instance \(37=6^{2}+1\), then \(a\) is even and \(m\) is a power of 2 .

Short Answer

Expert verified
For \(a^m+1\) to be prime, \(a\) must be even, and \(m\) a power of 2.

Step by step solution

01

Understand the Problem Statement

We need to show that if \(a \geq 2\) and \(a^m + 1\) is a prime number, then \(a\) must be even, and \(m\) must be a power of 2. A specific example given is \(37 = 6^2 + 1\). Our task is to generalize why these conditions hold.
02

Consider If \(a\) Is Odd

If \(a\) is odd, then \(a^m\) will also be odd for any integer \(m\). Adding 1 to an odd number results in an even number, \(a^m + 1\), which cannot be prime (unless it is 2, but \(a \geq 2\), and so \(a^m + 1 > 2\)). Thus, \(a\) cannot be odd; it must be even.
03

Consider Even Values of \(a\)

Assume \(a = 2k\), where \(k\) is an integer. In this case, \(a^m + 1\) becomes \((2k)^m + 1\). We need to consider under what conditions \((2k)^m + 1\) can be prime. Since \(2k\) is even, analyze \(m\) to determine when the expression remains prime.
04

Analyzing the Expression For Even \(m\)

If \(m\) is not a power of 2, for instance, if \(m = 2n \times k\) where \(k > 1\), then \((2k)^m\) can be factored using the identity of difference of squares or polynomials for larger powers. This makes \((2k)^m + 1\) a composite number, unless \(2n = 1\), meaning \(m\) is a power of 2.
05

Conclusion

Given steps 2 and 4, \(a\) must be even, and \(m\) must be a power of 2 to prevent \(a^m + 1\) from being expressible as a nontrivial product, hence remaining prime.

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Key Concepts

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

Prime Numbers
A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. Prime numbers play a crucial role in number theory due to their fundamental nature as the "building blocks" of all natural numbers. If a number can only be divided by 1 and itself without leaving a remainder, it's considered prime. For example:
  • The number 2 is the smallest and the only even prime number.
  • Numbers like 3, 5, and 7 are also prime, as they cannot be divided evenly by any numbers other than 1 and themselves.
Being able to recognize and understand prime numbers is key when working with problems involving their properties, as prime numbers form the basis for much of number theory and many mathematical concepts.
Even Numbers
Even numbers are integers divisible by 2 with no remainder. Their defining property is that they end in 0, 2, 4, 6, or 8 in their decimal notation. An even number can be expressed in the form of 2n, where n is an integer. Here are a few examples and their properties:
  • 2, 4, 6, and 8 are basic examples of even numbers.
  • Any number that can be split into two equal whole numbers is even, reinforcing its divisibility by 2.
Understanding whether a number is even is crucial when working with number properties such as divisibility and when relating to other mathematical concepts like exponents and power of two.
Exponents
Exponents are a way to express repeated multiplication of the same number. When a number is raised to an exponent, it is multiplied by itself the number of times indicated by the exponent. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, resulting in \(2 \times 2 \times 2 = 8\). Exponents are central in solving equations or simplifying expressions involving large numbers, such as:
  • Understanding that \(3^2 = 9\) shows repeated multiplication.
  • Exponents like \(a^m\) often arise in algebra and complex number expressions.
Exponents provide a compact way to express potentially large calculations, which is particularly important in formulas and in problems involving exponential growth or decay.
Power of Two
A power of two is a number that can be expressed as 2 raised to an exponent (e.g., \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\), etc.). These numbers include key properties that are crucial in fields like computer science and mathematics. The characteristics of powers of two are simple:
  • Each power of two doubles the previous one (e.g., \(2^3 = 8\) then \(2^4 = 16\)).
  • Powers of two are integral in binary systems, which are the foundation of digital computing.
Recognizing powers of two is especially useful when dealing with computational problems or algorithms, node calculations in networks, and so on. They illustrate the principle of exponential growth effectively.

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Most popular questions from this chapter

Show that \(F_{0} F_{1} \ldots F_{n-1}=F_{n}-2\) for all \(n \geq 1\)

Evaluate the Mersenne number \(M_{13}=2^{13}-1 .\) Is it prime?

For which primes \(p\) is \(p^{2}+2\) also prime?

Show that if \(n, q \geq 1\) then the number of multiples of \(q\) among \(1,2, \ldots, n\) is \(\lfloor n / q] .\) Hence show that if \(p\) is prime and \(p^{e} \| n !\), then \(e=\lfloor n / p\rfloor+\) \(\left\lfloor n / p^{2}\right]+\left\lfloor n / p^{3}\right\rfloor+\cdots\)

Are the following statements true or false, where \(a\) and \(b\) are positive integers and \(p\) is prime? In each case, give a proof or a counterexample: (a) If \(\operatorname{gcd}\left(a, p^{2}\right)=p\) then \(\operatorname{gcd}\left(a^{2}, p^{2}\right)=p^{2}\). (b) If \(\operatorname{gcd}\left(a, p^{2}\right)=p\) and \(\operatorname{gcd}\left(b, p^{2}\right)=p^{2}\) then \(\operatorname{gcd}\left(a b, p^{4}\right)=p^{3}\). (c) If \(\operatorname{gcd}\left(a, p^{2}\right)=p\) and \(\operatorname{gcd}\left(b, p^{2}\right)=p\) then \(\operatorname{gcd}\left(o b, p^{4}\right)=p^{2}\). (d) If \(\operatorname{gcd}\left(a, p^{2}\right)=p\) then \(\operatorname{gcd}\left(a+p, p^{2}\right)=p\).
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