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Prime Numbers - Advanced Concepts

Prime Numbers

A Prime Number is a positive integer divisible by no integers other than unity and itself.

Example: 2, 3, 5, 7, 11, etc.

Twin Primes

A pair of prime numbers that differ by 2 (successive odd numbers that are both Prime numbers).

Examples: (3,5), (5,7), (11,13), ...

It is not known whether the set of twin prime numbers ends or not.

Co-primes or Relatively prime numbers

A pair of numbers not having any common factors other than 1 or -1. (Or alternatively their greatest common factor is 1 or -1)

Example: 15 and 28 are co-prime, because the factors of 15 (1,3,5,15), and the factors of 28 (1,2,4,7,14,28) are not in common (except for 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).

Example: 6 (proper factors: 1,2,3) is a Perfect number because 1+2+3=6.

Example: 28 (proper factors: 1,2,4,7,14) is also a Perfect number, because 1+2+4+7+14=28.

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 you 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 problems.

Abundant numbers

Any positive integer that is less than the sum of its distinct proper factors (factors other than the number itself).

Example: 12 is abundant since its distinct proper factors are 1, 2, 3, 4, and 6 whose sum is 16.

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.

Example: 32 (=25) is a deficient number because the sum of its distinct proper factors is 31 (1+2+4+8+16).

Furthermore, numbers of the form pn are always deficient where p is a prime number and n is a postive integer.

Example: 35=243. The factors of 243 other than itself are 81, 27, 3 and 1.
Sum of these factors is 112, which is lesser than 243.
Similarly, 56=15625, the factors of 15625 are 1,5,25,125,625, and 3125. The sum of these is 3906 which is lesser than 15625.

Amicable numbers

A pair of integers, each of which is the sum of the distinct proper factors of the other.

Example: 220 and 284 are amicable numbers because:
  • Factors of 220 (other than itself) are 1,2,4,5,10,11,20,22,44,55,110.
    Sum of these factors = 284
  • Factors of 284 (other than itself) are 1,2,4,71,142.
    Sum of these factors = 220.

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, this assumption must therefore 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.