Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square . I need the answer step by step

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:

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:

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:

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: ,

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