Is this Really Prime?

First thing to keep in mind is that 2 primes as well as 3 primes when multiplied gives a composite number(non-prime). Moreover, a prime and a composite also gives a composite number unless the number next to prime is not 1.

Thus, we can just have 1 prime no. in the given three numbers.

Now. we can't take 0 as one of our numbers.

Now, we must take the two numbers as 1 and -1 cause these are the only two which will let our prime be a prime.

Now,

$(1) \times(-1) \times(x)$ = any prime number.

now the common difference between 1 and -1 is -2 or 2

, if it's 2 then the third number will be 3 and the product will yield us -3 (not a prime number)

thus, the common difference is -2, thus the third number is (-3).

now,

the three numbers are (-3), (-1) and (1).

Therefore,

$(1) \times(-1) \times(-3)$ =3

X = $(3+1-(-1)+(-3))^{3}$

X = $(2)^{3}$

$\huge{X = 8}$

$\huge{:P}$

• First, we must know any prime number only can be divided by 1 and itself . Let's use P for the prime number.
• Proof by contradiction : if those 3 numbers in arithmetic progression are integers , we can assume ERRONEOUSLY that one of the numbers is 1 and another of them is P .

• However, if we do this, the third number could only be a negative number. And, if a negative number multiplies 1 and P , the result will be below zero; in other words: not a prime.

• After that, we observe that we must have two negative integers, so the final result stay as positive prime integer P .

• It implies that one of the numbers still must be 1 , other one must be -P , and the other one must be -1 (so it multiplies with 1 and -P resulting in a positive prime integer).

• With this, we know there is a arithmetic progression with 2nd term -1 and 3rd term 1 . As we can easily see, the first term will be -3 . It implies that: $-P = -3 => P = 3$ .

• Afterwards, we only need to replace the three terms and the prime in $X = (P + a - b + c)^{P}$ .
  $X = [3 + (-3) - (-1) + 1]^{3}$ $X = 2^{3} = 8$ .
  

PS: i know the first part got a little messy, so you can ask anything and i'll try to help you out. I'm sorry, english is not my first language, and it is a really bit hard to develop an extense (and a little complex) induction with mathematic terms that I am still learning. Feel free to say anything you want about the solution. I'm here to help xD