Answer:
The family of all prime numbers such that
is a perfect square is represented by the following solution:
is an arbitrary prime number. (1)
(2)
is another arbitrary prime number. (3)
Step-by-step explanation:
From Algebra we know that a second order polynomial is a perfect square if and only if
. From statement, we must fulfill the following identity:
![a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}](https://tex.z-dn.net/?f=a%5E%7B2%7D%20%2B%20b%5E%7B2%7D%20%2B%20c%5E%7B2%7D%20-%201%20%3D%20x%5E%7B2%7D%20%2B%202%5Ccdot%20x%5Ccdot%20y%20%2B%20y%5E%7B2%7D)
By Associative and Commutative properties, we can reorganize the expression as follows:
(1)
Then, we have the following system of equations:
(2)
(3)
(4)
By (2) and (4) in (3), we have the following expression:
![(b^{2} - 1) = 2\cdot a \cdot c](https://tex.z-dn.net/?f=%28b%5E%7B2%7D%20-%201%29%20%3D%202%5Ccdot%20a%20%5Ccdot%20c)
![b^{2} = 1 + 2\cdot a \cdot c](https://tex.z-dn.net/?f=b%5E%7B2%7D%20%3D%201%20%2B%202%5Ccdot%20a%20%5Ccdot%20c)
![b = \sqrt{1 + 2\cdot a\cdot c}](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%7B1%20%2B%202%5Ccdot%20a%5Ccdot%20c%7D)
From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then,
. If
,
and
are prime numbers, then
must be an even composite number, which means that
and
can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.
In addition,
must be a natural number, which means that:
![1 + 2\cdot a\cdot c \ge 4](https://tex.z-dn.net/?f=1%20%2B%202%5Ccdot%20a%5Ccdot%20c%20%5Cge%204)
![2\cdot a \cdot c \ge 3](https://tex.z-dn.net/?f=2%5Ccdot%20a%20%5Ccdot%20c%20%5Cge%203)
![a\cdot c \ge \frac{3}{2}](https://tex.z-dn.net/?f=a%5Ccdot%20c%20%5Cge%20%5Cfrac%7B3%7D%7B2%7D)
But the lowest possible product made by two prime numbers is
. Hence,
.
The family of all prime numbers such that
is a perfect square is represented by the following solution:
is an arbitrary prime number. (1)
(2)
is another arbitrary prime number. (3)
Example:
, ![c = 2](https://tex.z-dn.net/?f=c%20%3D%202)
![b = \sqrt{1 + 2\cdot (2)\cdot (2)}](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%7B1%20%2B%202%5Ccdot%20%282%29%5Ccdot%20%282%29%7D)