Answer:
The family of all prime numbers such that [tex]a^{2} + b^{2} + c^{2} -1[/tex] is a perfect square is represented by the following solution:
[tex]a[/tex] is an arbitrary prime number. (1)
[tex]b = \sqrt{1 + 2\cdot a \cdot c}[/tex] (2)
[tex]c[/tex] 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 [tex](x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}[/tex]. From statement, we must fulfill the following identity:
[tex]a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}[/tex]
By Associative and Commutative properties, we can reorganize the expression as follows:
[tex]a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2}[/tex] (1)
Then, we have the following system of equations:
[tex]x = a[/tex] (2)
[tex](b^{2}-1) = 2\cdot x\cdot y[/tex] (3)
[tex]y = c[/tex] (4)
By (2) and (4) in (3), we have the following expression:
[tex](b^{2} - 1) = 2\cdot a \cdot c[/tex]
[tex]b^{2} = 1 + 2\cdot a \cdot c[/tex]
[tex]b = \sqrt{1 + 2\cdot a\cdot c}[/tex]
From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, [tex]a, b, c > 1[/tex]. If [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex] are prime numbers, then [tex]2\cdot a\cdot c[/tex] must be an even composite number, which means that [tex]a[/tex] and [tex]c[/tex] 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, [tex]b[/tex] must be a natural number, which means that:
[tex]1 + 2\cdot a\cdot c \ge 4[/tex]
[tex]2\cdot a \cdot c \ge 3[/tex]
[tex]a\cdot c \ge \frac{3}{2}[/tex]
But the lowest possible product made by two prime numbers is [tex]2^{2} = 4[/tex]. Hence, [tex]a\cdot c \ge 4[/tex].
The family of all prime numbers such that [tex]a^{2} + b^{2} + c^{2} -1[/tex] is a perfect square is represented by the following solution:
[tex]a[/tex] is an arbitrary prime number. (1)
[tex]b = \sqrt{1 + 2\cdot a \cdot c}[/tex] (2)
[tex]c[/tex] is another arbitrary prime number. (3)
Example: [tex]a = 2[/tex], [tex]c = 2[/tex]
[tex]b = \sqrt{1 + 2\cdot (2)\cdot (2)}[/tex]
[tex]b = 3[/tex]
