Prime Numbers and Euclids Proof
Prime numbers have been fascinating mathematicians for thousands of years.
A prime number is always greater than 1 and can only be divided by itself and 1 – no other number will divide into it. 2 is the first prime number (also the only even prime number), followed by 3, 5, 7, and so on.
Nonprime numbers are defined as composite numbers.
Prime numbers are distributed randomly throughout all the other numbers. Also, there is no simple and quick way to find a specific (new) prime number. As a result, large prime numbers are used in encryption to make the online platform protected for safe communication and financial transaction.
Did you know the number of primes is infinite? Explore Euclid’s Theorem of infinite primes with proof, solved examples and interactive questions, the Cuemath way.
Lesson PLan
How Do You Identify Prime Numbers?
Definition
In numbers, we have learnt that prime numbers have two factors  1 and the number itself. To find a prime number, we need to find the numbers which consist of only two factors. This is possible by using a simple method, which is called prime factorization.
Steps to Finding Prime Numbers Using Factorization
Step 1. Divide the number into factors
Step 2. Check the number of factors of that number. If the number of factors is more than 2 then it is composite.
Example: \(8\) has four factors \(1,\, 2,\, 4,\, 8\). So 8 and therefore is not prime
Step 3. All prime numbers greater than 3 can be represented by the formula \(6n+ 1\) and \(6n 1) for n greater than equal to 1. Exception if the number ends with 5 or higher multiples of 7 such as 49, 77, 91
Examples:
\(5=6\times 11\)
\(7=6\times 1+1\)
\(11=6\times 21\)
\(13=6\times 21\)
etc.
Steps to Identify A Large Number Is Prime Number
Lets us take a number \(1249\)
Step 1. If the number ends with \(0,\, 2,\, 4,\,5,\, 6\, and\, 8,\) then it is not a prime number.
Step 2. Add the digits of your number if the number is divisible by \(3\) then we can say that, it is not a prime number.
\(1249=1+2+4+9=16\)
Step 3. If the condition for steps 1 and 2 are not true, then find the square root of the number
\(\sqrt(1249)=35.3\)
Step 4. Divide the number by all prime numbers less than its square root value
Prime number less than \(35:\,\, 1,\,2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\,31\)
Step 5. If the number is divisible by any of the prime numbers less than its square root, it is not a prime number; otherwise, it is prime.
Hence, after following all steps we can say that \(1249\) is a prime number
What Is Euclid’s Proof for Infinite Primes?
Euclid, in 4th century B.C, pointed out that there has been an infinite number of primes. The concept of infinity is not known at that time. He said ”prime numbers are quite any fixed multitude of them” meaning if you imagine 100, there are more, and if you imagine a million there are still more.
Euclid's Proof for an Infinite Number of Primes
let us assume that, there are a finite number of primes, \(n\)
the largest prime number is \(p_n\).
Consider the number, \(N\). that is the product of these primes number plus one
By construction, N is not divisible by any of the \(p_i\).
Hence it's either prime itself, or divisible by another prime greater than \(p_n\) , contradicting the idea.
For example:
\(2 + 1 = 3\), is prime
\(2\times 3 + 1 = 7\), is prime
\(2\times 3 \times 5 + 1 = 31\), is prime
\(2\times 3 \times 5\times 7 + 1 = 211\), is prime
\(2\times 3 \times 5\times 7\times 11 + 1 = 2311\), is prime
etc.
Thus, we can conclude this the number of primes is infinite.
Solved Examples
Example 1 
Is \(233\) a prime number?
Solution
If \(233\) is a prime number, then it must not be divisible by a prime that is less than or equal to \(\sqrt{233}\).\(\sqrt{213}\) is between \(15\) and \(16\), so the largest prime number that is less than \(\sqrt{213}\) is \(13\). therefore we test \(2,\, 3,\, 5,\, 7,\, 11,\, and\, 13\,\) for divisibility.
233 is not divisible by any of those numbers, so it must be prime
\(233\) is a prime number 
Example 2 
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Is \(1001\) a prime number?
Solution
If \(1001\) is a prime number, then it must not be divisible by a prime that is less than or equal to \(\sqrt{1001}\).\(\sqrt{1001}\) is between \(31\) and \(32\), so the largest prime number that is less than \(\sqrt{1001}\) is \(31\). It is therefore sufficient to test \(2,\, 3,\, 5,\, 7,\, 11,\,13,\,17,\,19,\, 23,\, 29,\, and\, 31\) for divisibility.
\(1001=7 \times 11\times 13\). \(1001\) is divisible by \(7,\,11,\, and \, 13\) so it is not a prime number
\(1001\) is not a prime number 
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

\(1\) is the only whole number that we don't consider as prime nor composite.

The lowest even prime number is \(2\) and is also the only even prime number.

The lowest odd prime number is \(3\).

Most common pattern in all prime numbers is except \(2\) and \(5\) ends in \(1,\, 3,\, 7,\, or\, 9\).

If \(N\) is composite, then it must be divisible by any prime number \(p\) such that \(p<=\sqrt{N}\).
Let's Summarize
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FAQs on Prime Numbers And Euclid's Proof
1. What did Euclid prove about prime numbers in 300 BC?
Euclid, in 4th century B.C, points out that there have been an infinite Primes. The concept of infinity is not known at that time. He said ”prime numbers are quite any fixed multitude of them” meaning if you imagine 100, there are more, and if you imagine a million there are still more.
2. Are twin primes infinite?
"Twin primes" are primes that are two steps apart from each other on that line: 3 and 5, 5 and 7, 29 and 31, 137 and 139, and so on. The twin prime conjecture states that there are infinitely many twin primes, and that you'll keep encountering them no matter how far down the number line you go.
3. Is 1 a Prime Number?
There are people that believe so because they say that 1 can only be divided by 1 and itself but in mathematics, the number 1 has been discarded as a prime number because it only has one divisor. In fact, the criteria of ”a positive integer is prime if it has exactly two positive divisors” is used to exclude the number one from the prime number list. It is not because we are being picky about it, but if the number one was considered prime then many mathematical properties would have to be said differently.
4. Is 0 finite or infinite?
The concept of zero and that of infinity are linked, but, obviously, zero is not infinity. Rather, if we have N / Z, with any positive N, the quotient grows without limit as Z approaches 0. Hence we readily say that N / 0 is infinite.
5. What is meant by the Euclidean Geometry
Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 BCE). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools
6. Is there a pattern to prime numbers?
A clear rule determines exactly what makes a prime: it's a whole number that can't be exactly divided by anything except 1 and itself. But there's no discernable pattern in the occurrence of the primes. ... That's because after the number 5, there are only four possibilities — 1, 3, 7 and 9 — for prime last digits.
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