Twin primes (a) let p > 3 be a prime. prove that p is of the form 3k +1 or 3k – 1 for some integer k. (b) twin primes are pairs of prime numbers p and q that have a difference of 2. use part (a) to prove that 5 is the only prime number that takes part in two different twin prime pairs.

(a) Let us recall the division algorithm: given two positive integers and there exist other two positive integers and such that

where and is called the remainder.

So, given any positive integer and 3 we can write

where . Thus, every can be written as

Now, notice that . Hence, every number can be written as , or or .

A number is prime if and only if its only factors are 1 and itself. So, a number of the form cannot be prime. Therefore, every primer number is of the form or .

(b) Assume that there are three prime numbers such that , and are prime.

By the previous exercise or . Let us analyze both cases separately.

First case: . Then that can be prime, and that is not prime. Hence, there are not such three primes with .

Second case: . Then, that can be prime, and that cannot be prime. Hence, there are not such three primes with .

Therefore, there are no three primes  of the form , and , except for 3, 5 and 7.

Notice that this is only possible because 5=2*3-1 and 2*3-3=3, that is the only ‘‘multiple’’ of 3 that is prime.

80 the answer is 80

step-by-step explanation:

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.