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Chapter 4: Problem 33

(a) Prove that every odd positive integer of the form \(3 n+2, n \in N\), has a prime factor of the same form. What happens if the word odd is omitted? (b) Repeat (a) for positive integers of the form \(4 n+3\). (c) Repeat (a) for positive integers of the form \(6 n+5\). (d) Prove that there are infinitely many primes of the form \(6 n+5\).

Short Answer

Expert verified
Numbers of form '3n+2', '4n+3', and '6n+5' must have primes of the same form. There are infinitely many primes of form '6n+5'.

Step by step solution

01

Understanding the form 3n+2 with odd rule

Every number of the form \(3n+2\) is not divisible by 3. To ensure it is odd, \(n\) must be odd, because if \(n\) were even, \(3n+2\) would be odd itself. Therefore, if \(3n+2\) is odd and not divisible by 3, it must have a prime factor also not divisible by 3, which would be of the form \(3k+2\).
02

Understanding 3n+2 without the odd restriction

If the number is of the form \(3n+2\) without requiring it to be odd, then \(n\) can be either odd or even, having a prime factor of the same form. For any \(3n+2\), it cannot have all prime factors of the form \(3k+1\) since multiplying such factors wouldn鈥檛 result in \(3n+2\).
03

Repeating for 4n+3 (Part b)

For numbers of the form \(4n+3\), observe that they are not divisible by 4. If all prime factors were of the form \(4k+1\), the product would remain in that form. Therefore, it must have at least one prime factor of the form \(4k+3\). If the word 'odd' is omitted, the requirement automatically holds as it cannot be of form \(4k\) due to residue issues.
04

Repeating for 6n+5 (Part c)

For numbers of the form \(6n+5\), they are not divisible by 6. A prime factor must be of a form like \(6k+5\) since any factor of the form \(6k+1\) cannot alone construct \(6n+5\).
05

Prove infinitely many primes of 6n+5 (Part d)

Construct numbers of form \(6k+5\). If assumed finite, take their product plus one to contradict the set of forms starting anew with a number modulo 6 not zero, specifically as \(6k+5\). There cannot be a complete composite system without prime generation in that form, ensuring the persistence of prime forms \(6n+5\).

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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 the building blocks of mathematics. They are defined as natural numbers greater than 1 that have no divisors other than 1 and themselves.
Prime numbers possess unique properties that play a crucial role in number theory and various branches of mathematics.
  • A prime number must be indivisible by any other number except for 1 and itself.
  • The smallest prime number is 2, which is also the only even prime number. All other primes are odd.
  • Examples of prime numbers include 3, 5, 7, 11, 13, etc.
Prime numbers are essential for understanding the concepts of factorization and number forms, as discussed in the exercise. Discovering the primality of numbers in specific forms helps unleash deeper mathematical patterns.
Number Forms
Number forms refer to specific algebraic expressions under which numbers are categorized. In problems concerning number theory, particular number forms help in exploring deeper properties and factors.
For example:
  • The form \(3n+2\), where \(n\) is an integer, represents a sequence of numbers like 2, 5, 8, and so on.
  • Each form can exhibit unique arithmetic properties. For instance, numbers of the form \(3n+2\) are not divisible by 3, highlighting a key feature in recognizing patterns of divisibility and factorization.
  • Understanding number forms like \(4n+3\) or \(6n+5\) also plays a significant role in identifying special properties for these sequences and their use of primes.
Recognizing these forms enables mathematicians to determine if the numbers have distinct prime factors, forming the core of many proof-related exercises.
Factorization
Factorization involves breaking down numbers into their prime components. It鈥檚 a fundamental concept in number theory representing the decomposition of a number into a product of simpler elements鈥攊ts prime factors.
  • Each number greater than 1 can be expressed uniquely as a product of primes, a concept known as the Fundamental Theorem of Arithmetic.
  • For odd positive integers like those in the form \(3n+2\), finding at least one prime factor in the same form ensures specific properties about divisibility and compatibility with these forms.
  • In the exercise, proving that numbers like \(4n+3\) or \(6n+5\) similarly have particular prime factors is tied to the certainty of factorization patterns that stay consistent within these modular forms.
Through factorization, it鈥檚 seen how unique number forms necessitate certain primes, lending itself to greater exploration of number properties.
Infinitude of Primes
The infinitude of primes is a profound discovery, denoting that there are endless prime numbers. This ensures that primes exist in any number form, infinitely.
Euclid's classic proof shows that assuming a finite set of primes leads to contradictions. The exercise extends this idea, particularly aiming to show infinite primes in specific forms.
  • To prove this for numbers like \(6n+5\), mathematicians demonstrate by creating new numbers鈥攆ormulated from known primes plus one鈥攁s they tend not to be divisible by previously recognized primes.
  • The addition of 1 forces the existence of a new prime factor outside the assumed finite set, proving the infinite nature of primes in that form.
  • Such proofs underline the power and necessity of infinite primes, solidifying their role and presence across numerous mathematical constructs and number forms.
Understanding the infinitude of primes reveals insights into the consistency and structure of mathematical systems, reassuring that not only are primes unbounded, but they also permeate every conceivable number form.

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Most popular questions from this chapter

In each of the following, the given integer \(p\) is a prime. (a) \([\mathrm{BB}]\) Find \(18^{8970}, 18^{8971}\) and \(18^{8972}\), each mod \(p\), \(p=8971\) (b) Find \(53^{20.592}, 53^{20,593}, 53^{20.594}\), each mod \(p, p=\) 20,593 (c) Find \(3^{508}, 3^{509}, 3^{512}\), each mod \(p, p=509\) (d) Find \(6^{10.588}, 6^{10.589}, 6^{10.594}\), each mod \(p, p=\) 10,589 (e) Find \(8^{4056}, 8^{4058}, 8^{4060}\), each mod \(p, p=4057\). (f) Find \(2^{3948}\) and \(2^{3941}(\bmod p), p=3943\).

True or false: \(\left\\{n \in \mathrm{N} \mid n>2\right.\) and \(a^{n}+b^{n}=\) \(c^{n}\) for some \(\left.a, b, c \in N\right\\}=\emptyset ?\)

Let \(\mathrm{Q}^{+}\) denote the set of positive rational numbers. If \(\frac{m}{n}\) is in \(\mathrm{Q}^{+}\), then \- either \(m=1\) or \(m=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{k}^{e_{k}}\) is the product of powers of distinct primes \(p_{1}, p_{2}, \ldots, p_{k}\) \- either \(n=1\) or \(n=q_{1}^{f_{1}} q_{2}^{f_{2}} \cdots q_{\ell}^{f_{t}}\) is the product of powers of distinct primes \(q_{1}, q_{2}, \ldots, q_{\ell}\) and, in the case \(m \neq 1\) and \(n \neq 1\), we may assume that no \(p_{i}\) equals any \(q_{j}\), Now define \(f: \mathrm{Q}^{+} \rightarrow \mathrm{N}\) by \(f(1)=1\) \(f\left(\frac{m}{n}\right)=\left\\{\begin{array}{ll}p_{1}^{2 e_{1}} p_{2}^{2 e_{2}} \cdots p_{k}^{2 e_{k}} & \text { if } n=1, m \neq 1 \\ q_{1}^{2 f_{1}-1} q_{2}^{2 f_{2}-1} \cdots q_{\ell}^{2 f_{\ell}-1} & \text { if } m=1, n \neq 1 \\ p_{1}^{2 e_{1}} \cdots p_{k}^{2 e_{k}} q_{1}^{2 f_{1}-1} & \cdots q_{\ell}^{2 f_{\ell}-1} & \text { if } m \neq 1, n \neq 1\end{array}\right.\) (a) Find \(f(8)[\mathrm{BB}], f\left(\frac{1}{8}\right), f(100)\) and \(f\left(\frac{40}{63}\right)\). (b) Find \(x\) such that \(f(x)=1,000,000[\mathrm{BB}], t\) such that \(f(t)=10,000,000\) and \(s\) such that such that \(f(s)=365,040\) (c) Show that \(f\) is a bijection. (This exercise, which gives a direct proof that the positive rationals are countable, is due to Yoram Sagher. See the American Mathematical Monthly, Vol. 96 (1989), no. 9, p. \(823 .\) )

If \(a\) and \(b\) are relatively prime integers, prove that \(\operatorname{gcd}(a+b, a-b)=1\) or 2

Find \(q\) and \(r\) as defined by the division algorithm, \(4.1 .5\), in each of the following cases. (a) \(a=5286 ; b=19\) (b) \(a=-5286 ; b=19\) (c) \(a=5286 ; b=-19\) (d) \(a=-5286 ; b=-19\) (e) \(a=19 ; b=5286\) (f) \(a=-19 ; b=5286\)
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