What Is A Prime Number
A prime number is a number that can only be divided by itself and 1 without remainders. Here we explain what exactly this means, give you a list of the prime numbers children need to know at primary school and provide you with some practice prime number questions and examples.
what is a prime number
Greek mathematician Euclid (one of the most famous mathematicians of the classical era), recorded a proof that there is no largest prime number among the set of primes. However, many scientists and mathematicians are still searching to find it as part of the Great Internet Mersenne Prime Search.
By multiplying two very large prime numbers together (some companies use prime numbers that are hundreds of digits long!), we create an even larger number whose original factors (the two very large prime numbers) are only known to us. We then use this even larger number to encrypt our information.
If anyone else wants to discover what information we are sending, they have to find out what our original factors were. With prime numbers as long as the ones we have used, it could take them years or even decades of constant trial and error before they find even one. This kind of public-key cryptography ensures our information is kept safe.
CHALLENGE QUESTION: Chen chooses a prime number. He multiplies it by 10 and then rounds it to the nearest hundred. His answer is 400. Write all the possible prime numbers Chen could have chosen.
Prime numbers are the numbers that have only two factors, that are, 1 and the number itself. Consider an example of number 5, which has only two factors 1 and 5. This means it is a prime number. Let us take another example of the number 6, which has more than two factors, i.e., 1, 2, 3, and 6. This means 6 is not a prime number. Now, if we take the example of the number 1, we know that it has only one factor. So, it cannot be a prime number as a prime number should have exactly two factors. This means 1 is neither a prime nor a composite number, it is a unique number.
A number greater than 1 with exactly two factors, i.e., 1 and the number itself is a prime number. For example, 7 has only 2 factors, 1 and 7 itself. So, it is a prime number. However, 6 has four factors, 1, 2, 3 and 6. Therefore, it is not a prime number. It is a composite number.
Method: Every prime number, apart from 2 and 3, can be written in the form of '6n + 1 or 6n - 1'. So, if we have any number different from 2 and 3, we can check if it is prime or not by trying to express it in the form of 6n + 1 or 6n - 1
A prime number chart is a chart that shows the list of prime numbers in a systematic order. It should be noted that all prime numbers are odd numbers except for the number 2 which is an even number. Interestingly, 2 is the only prime number that is even. This means the list of odd numbers can start from 3 onwards and continue because the rest of the prime numbers are odd numbers. For example, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and so on are odd prime numbers.
Prime numbers have created human curiosity since ancient times. Even today, mathematicians are trying to find prime numbers with mystical properties. Euclid proposed the theorem on prime numbers - there are infinitely many prime numbers.
Do you know all the prime numbers from 1 to 100? Did you check if each number is divisible by the smaller numbers? Eratosthenes was one of the greatest scientists who lived a few decades after Euclid. He designed a smart way to determine all the prime numbers up to a given number. This method is called the Sieve of Eratosthenes. Let us learn about the Sieve of Eratosthenes in the following section.
The Sieve of Eratosthenes is an ancient algorithm that helps to find prime numbers up to any given limit. The following figure shows the prime numbers up to 100 that are found using the Sieve of Eratosthenes. The uncrossed numbers in the figure represent the prime numbers that are left after using the Sieve of Eratosthenes.
Suppose we need to find the prime numbers up to 'n', so we will generate the list of all numbers from 2 to n. The following steps show how to find all the prime numbers up to 100 using the Sieve of Eratosthenes.
The smallest prime number is 2. Interestingly, this is the only even number that is a prime number. It should be noted that 1 is neither a prime number nor a composite number because it has only 1 factor which is 1, hence, it is a unique number.
The number 15 has more than two factors: 1, 3, 5, and 15. Hence, it is a composite number. On the other hand, 13 has just two factors: 1 and 13. Hence, it is a prime number. Therefore, 13 is a prime number.
Prime numbers are those numbers that have only two factors, i.e., 1 and the number itself. For example, 2, 3, 7, 11, and so on are prime numbers. On the other hand, numbers with more than 2 factors are called composite numbers.
One of the easiest methods to find that a given number p, is a prime number, is to check the number of factors of the number p. If p has exactly two factors, 1 and p, then we say that p is a prime number.
A prime number is a number that has only two factors, that is, 1 and the number itself. For example, 2, 3, 5, 7 are prime numbers. Co-prime numbers are the set of numbers whose Highest Common Factor (HCF) is 1. For example, 2 and 3 are co-prime numbers.
Twin prime numbers are those pairs of prime numbers that have a difference of 2 between them. For example, 3 and 5 are twin prime numbers because 5 - 3 = 2. These pairs of numbers always have one composite number between them. In this case, 4 is a composite number that comes in between 3 and 5. Other examples of twin prime numbers are (5, 7), (11, 13) and so on.
The numbers that are not prime numbers are called composite numbers. Composite numbers are those numbers that have more than 2 factors. For example, 4 is a composite number because it has three factors, 1, 2, and 4. Similarly, 44 is a composite number because it has six factors, 1, 2, 4, 11, 22 and 44.
2 is the only even prime number from 1 to 100. In fact, 2 is the only even number that is prime. All other even numbers are composite numbers because they have more than 2 factors. For example, the factors of 4 = 1, 2, and 4, Similarly the factors of 6 = 1, 2, 3, 6. therefore there are no even prime numbers except 2.
The largest prime number discovered so far is 2 raised to the 57,885,161st power minus 1, or 257,885,161 - 1. It is 17,425,170 digits long. It was discovered by University of Central Missouri mathematician Curtis Cooper as part of a giant network of volunteer computers devoted to finding primes.
In 200 B.C., Eratosthenes created an algorithm that calculated prime numbers, known as the Sieve of Eratosthenes. This algorithm is one of the earliest algorithms ever written. Eratosthenes put numbers in a grid, and then crossed out all multiples of numbers until the square root of the largest number in the grid is crossed out. For example, with a grid of 1 to 100, you would cross out the multiples of 2, 3, 4, 5, 6, 7, 8, 9, and 10, since 10 is the square root of 100. Since 6, 8, 9 and 10 are multiples of other numbers, you no longer need to worry about those multiples. So for this chart, you would cross out the multiples of 2, 3, 5 and 7. With these multiples crossed out, the only numbers that remain and are not crossed out are prime. This sieve enables someone to come up with large quantities of prime numbers.
But during the Dark Ages, when intellect and science were suppressed, no further work was done with prime numbers. In the 17th century, mathematicians like Fermat, Euler and Gauss began to examine the patterns that exist within prime numbers. The conjectures and theories put out by mathematicians at the time revolutionized math, and some have yet to be proven to this day. In fact, proof of the Riemann Hypothesis, based on Bernhard Riemann's theory about patterns in prime numbers, carries a $1 million prize from the Clay Mathematics Institute. [Related: Famous Prime Number Conjecture One Step Closer to Proof]
In 1978, three researchers discovered a way to scramble and unscramble coded messages using prime numbers. This early form of encryption paved the way for Internet security, putting prime numbers at the heart of electronic commerce. Public-key cryptography, or RSA encryption, has simplified secure transactions of all times. The security of this type of cryptography relies on the difficulty of factoring large composite numbers, which is the product of two large prime numbers.
Confidence in modern banking and commerce systems hinges on the assumption that large composite numbers cannot be factored in a short amount of time. Two primes are considered as sufficiently secure if they are 2,048 bits long, because the product of these two primes would be about 1,234 decimal digits.
Prime numbers even show up in nature. Cicadas spend most of their time hiding, only reappearing to mate every 13 or 17 years. Why this specific number? Scientists theorize that cicadas reproduce in cycles that minimize possible interactions with predators. Any predator reproductive cycle that divides the cicada's cycle evenly means that the predator will hatch out the same time as the cicada at some point. For example, if the cicada evolved towards a 12-year reproductive cycle, predators who reproduce at the 2, 3, 4 and 6 year intervals would find themselves with plenty of cicadas to eat. By using a reproductive cycle with a prime number of years, cicadas would be able to minimize contact with predators.
This may sound implausible (obviously, cicadas don't know math), but simulation models of 1,000 years of cicada evolution prove that there is a major advantage for reproductive cycle times based on primes. It can be viewed here at _numbers/. It may not be intentional on the part of Mother Nature, but prime numbers show up more in nature and our surrounding world than we may think. 041b061a72