A Prime Number can be divided evenly only by 1 or itself. And it must be a whole number greater than 1.
Twin Primes
A pair of prime numbers that differ by 2 (successive odd numbers that are both Prime numbers).
It is not known whether the set of twin prime numbers ends or not.
Coprimes or Relatively prime numbers
A pair of numbers not having any common factors other than 1 or -1. (Or alternatively theirgreatest common factor is 1 or -1)
Mersenne's Primes
Prime numbers of the form 2n-1 where n must itself be prime.
3, 7, 31, 127 etc. are Mersenne primes.
Not all such numbers are primes. For example, 2047 (i.e. 211-1) is not a prime number. It is divisible by 23 and 89.
Mersenne's Primes are named after the French monk, theologian, philosopher and number-theorist Marin Mersenne (1588-1648 AD).
Perfect numbers
Any positive integer that is equal to the sum of its distinct proper factors (factors other than the number itself).
Euclid proved that 2n-1(2n-1) is an even perfect number when 2n-1 is a Mersenne prime. These are now called Euclid numbers and Euler proved that all even Perfect numbers are of this form for some positive prime number n. Thus, 6, 28, 496 are Perfect and correspond to values of 3, 7, and 31 for 2n-1 in the formula.
This table shows the results for n=1 to 13 which include the first five Perfect numbers:
| n | 2n-1 | 2n-1(2n-1) | Perfect? | Comment |
|---|---|---|---|---|
| 1 | 1 | 1 | No | n is not prime |
| 2 | 3 | 6 | Yes | n is prime, 2n-1 is prime |
| 3 | 7 | 28 | Yes | n is prime, 2n-1 is prime |
| 4 | 15 | 120 | No | n is not prime |
| 5 | 31 | 496 | Yes | n is prime, 2n-1 is prime |
| 6 | 63 | 2016 | No | n is not prime |
| 7 | 127 | 8128 | Yes | n is prime, 2n-1 is prime |
| 8 to 10 | ... | ... | No | not prime |
| 11 | 2047 | 2096128 | No | n is prime, but 2n-1 is not prime |
| 12 | 4095 | 8386560 | No | n is not prime |
| 13 | 8191 | 33550336 | Yes | n is prime, 2n-1 is prime |
Whether there are infinitely many even Perfect numbers or any odd perfect numbers remain unsolved questions.
Abundant numbers
Any positive integer that is less than the sum of its distinct proper factors (factors other than the number itself).
Deficient Numbers
Any positive integer that exceeds the sum of its distinct proper factors.
Any prime number is deficient, because it has only one proper factor: 1.
All numbers of the form 2n are also deficient.
Furthermore, numbers of the form pn are always deficient where p is a prime number and n is a positive integer.
Amicable numbers
A pair of integers, each of which is the sum of the distinct proper factors of the other.
Euclid's proof that the set of prime numbers is endless
The proof works by showing that if we assume that there is a biggest prime number, then there is a contradiction.
We can number all the primes in ascending order, so that P1 = 2, P2 = 3, P3 = 5 and so on. If we assume that there are just n primes, then the biggest prime will be labelled Pn . Now we can form the number Q by multiplying together all these primes and adding 1, so
Q = (P1 × P2 × P3 × P4... × Pn) + 1
Now we can see that if we divide Q by any of our n primes there is always a remainder of 1, so Q is not divisible by any of the primes.
But we know that all positive integers are either primes or can be decomposed into a product of primes. This means that either Q must be prime or Q must be divisible by primes that are larger than Pn.
Our assumption that Pn is the biggest prime has led us to a contradiction, so this assumption must be false, so there is no biggest prime.
Goldbach's Conjecture
The conjecture that every even number (greater than or equal to 6) can be written as the sum of two odd prime numbers.
Goldbach's Conjecture is named after Prussian born number-theorist and analyst Christian Goldbach (1690-1764 AD) who was a Professor of Mathematics at, and the Historian of, the Russian Imperial Academy. He was also the tutor of Peter the Great, and was a member of the Tsar's foreign ministry.
Goldbach also conjectured that all odd numbers are the sum of three odd primes: Vinogradov's theorem shows this true of all except possibly finitely many odd numbers.
Definition
Two numbers are said to be Coprimes when they have only 1 as a common factor.
Following examples would further help you to understand Coprimes
Example 1 = Are numbers 6 and 25 Coprime ?
Answer = Two given numbers are 6 and 25
Factors of 6 = 1, 2, 3, 6
Factors of 25 = 1, 5, 25
On comparing the factors of numbers 6 and 25, you can see that both have only 1 as a common factor,
Hence we can say that numbers 6 and 25 are Coprimes.
Example 2 = Are numbers 12 and 21 Coprime ?
Answer = Two given numbers are 12 and 21
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 21 = 1, 3, 7, 21
On comparing the factors of numbers 12 and 21, you can see that both have 1 and 3 as common factors,
Hence, we can say that numbers 12 and 21 are not Coprimes.
Example 3 = Are numbers 27 and 16 Coprime ?
Answer = Two given numbers are 27 and 16
Factors of 27 = 1, 3, 9, 27
Factors of 16 = 1, 2, 4, 16
On comparing the factors of numbers 27 and 16, you can see that both have 1 as a common factor,
Hence we can say that numbers 27 and 16 are Coprimes.
Definition
Two numbers are said to be Coprimes when they have only 1 as a common factor.
Following examples would further help you to understand Coprimes
Example 1 = Are numbers 6 and 25 Coprime ?
Answer = Two given numbers are 6 and 25
Factors of 6 = 1, 2, 3, 6
Factors of 25 = 1, 5, 25
On comparing the factors of numbers 6 and 25, you can see that both have only 1 as a common factor,
Hence we can say that numbers 6 and 25 are Coprimes.
Example 2 = Are numbers 12 and 21 Coprime ?
Answer = Two given numbers are 12 and 21
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 21 = 1, 3, 7, 21
On comparing the factors of numbers 12 and 21, you can see that both have 1 and 3 as common factors,
Hence, we can say that numbers 12 and 21 are not Coprimes.
Example 3 = Are numbers 27 and 16 Coprime ?
Answer = Two given numbers are 27 and 16
Factors of 27 = 1, 3, 9, 27
Factors of 16 = 1, 2, 4, 16
On comparing the factors of numbers 27 and 16, you can see that both have 1 as a common factor,
Hence we can say that numbers 27 and 16 are Coprimes.
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