Question

Twin Primes Conjecture A natural number is called a prime number if it has exactly two factors, 1 and itself. 1 is not a prime number because it has exactly one factor. If a number is not prime it is called composi -List the numbers 1 through 31 and circle the primes. What do you see? -You might notice that some pairs of prime numbers have exactly one composite number between them. Such pairs of prime numbers include 3&5, 11&13, 17&19, 29&31. These pair numbers are called twin primes. -Write your first impression regarding this question: Are there an infinite number of twin primes? Provide a justification for your thinking. -Do a little research on the twin prime conjecture and describe at least one interesting fact ti you find.

Answer 1

Answer:

See explanation below.

Step-by-step explanation:

The prime numbers are bold:

1  2  3  4  5  6  7  8  9  10  11 12  13  14  15  16  17

18  19  20  21  22  23  24  25  26  27  28  29  30  31

a) We can see that as we go higher, twin primes seem less frequent but even considering that, there is an infinite number of twin primes. If you go high enough you will still eventually find a prime that is separated from the next prime number by just one composite number.

b) I think it's interesting the amount of time that has been devoted to prove this conjecture and the amount of mathematicians who have been involved in this. One of the most interesting facts was that in 2004 a purported proof (by R. F. Arenstorf) of the conjecture was published but a serious error was found on it so the conjecture remains open.