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Chapter 8: Problem 14

(a) Determine five different primes that are congruent to 3 modulo 4 . (b) Prove that there are infinitely many primes that are congruent to 3 modulo \(4 .\)

Short Answer

Expert verified
(a) Five different primes congruent to 3 modulo 4 are 3, 7, 11, 19, and 23. (b) By using proof by contradiction and assuming there are only finitely many primes congruent to 3 modulo 4, we can derive a contradiction, demonstrating that there are indeed infinitely many primes congruent to 3 modulo 4.

Step by step solution

01

Part (a): Find five prime numbers congruent to 3 modulo 4

To find primes congruent to 3 modulo 4, we can list down the numbers that are congruent to 3 modulo 4 and then see which are prime. The general form of these numbers is \(4n + 3\), where n is a non-negative integer. The first few numbers in this form are: 3, 7, 11, 15, 19, 23, 27, 31... Now, we can see that 3, 7, 11, 19, and 23 are all prime numbers, so these are our five different primes congruent to 3 modulo 4.
02

Part (b): Proof by contradiction for infinitely many primes congruent to 3 modulo 4

To prove that there are infinitely many primes that are congruent to 3 modulo 4, we can use a proof by contradiction. We will assume that there are only finitely many primes congruent to 3 modulo 4 and derive a contradiction. Let's consider a finite set of primes, say \(P = \{p_1, p_2, ..., p_n\}\), where all primes are congruent to 3 modulo 4. Let's construct a new number, Q, using these primes as follows: \[Q = (4p_1p_2...p_n) - 1\] Now, by construction, Q is congruent to 3 modulo 4. Let R be a prime divisor of Q. Since Q is congruent to 3 modulo 4, R must also be congruent to 3 modulo 4. If R is not in our set P, then we have found a new prime congruent to 3 modulo 4, which contradicts our assumption that P contains all such primes. If R is in our set P, let's say R = \(p_k\), then (as per the construction of Q), \(Q = 4p_1p_2...p_n - 1\) is divisible by \(p_k\). But \(4p_1p_2...p_n\) is also divisible by \(p_k\). Then, by definition, their difference, 1, must be divisible by \(p_k\), which is not possible since 1 is not divisible by any prime number. In both cases, we get a contradiction. Therefore, our assumption that there are only finitely many primes congruent to 3 modulo 4 must be false. This proves that there are infinitely many primes that are congruent to 3 modulo 4.

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Key Concepts

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

Proof by Contradiction
Proof by contradiction is a fundamental method in mathematics used to establish the truth of a statement. The technique involves assuming the opposite of what one wants to prove and then showing that this assumption leads to an inconsistency or contradiction. In essence, since the assumption that the statement is false leads to an absurdity, the original statement is concluded to be true.

For instance, in proving there are infinitely many primes congruent to 3 modulo 4, we first assume the opposite 鈥� that there is a finite number of such primes. We then demonstrate that this assumption logically leads to an impossible situation, thus confirming that there must be infinitely many such primes. This method is particularly powerful as it often allows mathematicians to demonstrate the veracity of statements without direct construction or enumeration.
Congruence in Number Theory
Congruence in number theory is a way of stating that two numbers have the same remainder when divided by a given number, referred to as the modulus. When we say that two numbers are congruent modulo a certain number, we imply that their difference is divisible by that number.

Considering our prime example, any number that is congruent to 3 modulo 4 can be expressed as the form 4n + 3, where n is an integer. This means when you divide such a number by 4, the remainder is always 3. This concept is a cornerstone in number theory and is used in a wide array of proofs and theorems. In the context of primes, understanding congruence helps us categorize prime numbers into different classes based on their properties concerning moduli.
Infinite Prime Numbers
The concept that there are infinite prime numbers is an ancient discovery in mathematics, with roots tracing back to Euclid. A prime number is a natural number that is greater than 1 and cannot be formed by multiplying two smaller natural numbers. The fundamental theorem of arithmetic states that each natural number greater than 1 is either a prime itself or can be factored into primes, which are sometimes referred to as the 'building blocks' of natural numbers.

Primes are infinite in count as proven using various methods such as the classic proof by Euclid and, for our specific case, the proof showing there is an infinite number of primes congruent to 3 modulo 4. By proving statements regarding the infinitude of primes in various forms, we expand our understanding of number theory, allowing us to further delve into the mysteries of numbers and their inherent patterns.

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Most popular questions from this chapter

The Twin Prime Conjecture states that there are infinitely many twin primes, but it is not known if this conjecture is true or false. The answers to the following questions, however, can be determined. (a) How many pairs of primes \(p\) and \(q\) exist where \(q-p=3\) ? That is, how many pairs of primes are there that differ by 3 ? Prove that your answer is correct. (One such pair is 2 and \(5 .\) ) (b) How many triplets of primes of the form \(p, p+2,\) and \(p+4\) are there? That is, how many triplets of primes exist where each prime is 2 more than the preceding prime? Prove that your answer is correct. Notice that one such triplet is \(3,5,\) and 7

Let \(a, b,\) and \(c\) be integers with \(a \neq 0\) and \(b \neq 0,\) and let \(d=\operatorname{gcd}(a, b)\). If \(d\) does not divide \(c\), then the linear Diophantine equation \(a x+b y=c\) has no solution.

Is the following proposition true or false? Justify your conclusion. $$ \text { If } n \in \mathbb{N}, \text { then } \operatorname{gcd}(5 n+2,12 n+5)=1 \text { . } $$

(a) Find integers \(u\) and \(v\) such that \(9 u+14 v=1\) or explain why it is not possible to do so. Then find integers \(x\) and \(y\) such that \(9 x+14 y=10\) or explain why it is not possible to do so. (b) Find integers \(x\) and \(y\) such that \(9 x+15 y=10\) or explain why it is not possible to do so. (c) Find integers \(x\) and \(y\) such that \(9 x+15 y=3162\) or explain why it is not possible to do so.

(a) Let \(a\) and \(b\) be nonzero integers. If there exist integers \(x\) and \(y\) such that \(a x+b y=1,\) what conclusion can be made about \(\operatorname{gcd}(a, b) ?\) Explain. (b) Let \(a\) and \(b\) be nonzero integers. If there exist integers \(x\) and \(y\) such that \(a x+b y=2,\) what conclusion can be made about \(\operatorname{gcd}(a, b) ?\) Explain.
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