How to Find the Sum of Prime Numbers Satisfying Specific Conditions

In this article, we will explore the mathematical process of finding prime numbers that satisfy specific conditions. The objective is to find all prime numbers ( p ) such that both (frac{p 1}{2}) and (frac{p^2-1}{2}) are perfect squares. We will walk through the mathematical steps to determine the solution and explain the logic behind each step.

Mathematical Setup and Conditions

Let's denote [ p 2n^2 - 1 ] for some integer ( n ). This equation stems from the condition that (frac{p 1}{2}) is a perfect square. From this, we have:

[ frac{p 1}{2} n^2 ]

Substituting ( p 2n^2 - 1 ) into the second condition (frac{p^2-1}{2}) being a perfect square, we get:

[ frac{p^2-1}{2} frac{(2n^2 - 1)^2 - 1}{2} frac{4n^4 - 4n^2 1 - 1}{2} 2n^4 - 2n^2 ]

This expression needs to be a perfect square, so we set:

[ 2n^4 - 2n^2 - m^2 0 ]

Solving the Quadratic Equation

Let [ x n^2 ], the equation transforms into:

[ 2x^2 - 2x - m^2 0 ]

The discriminant of this quadratic equation must be a perfect square for [ x ] to be an integer. The discriminant is:

[ D 4 - 8m^2 4(m^2 - 1) ]

Setting [ D k^2 ], we get:

[ 4(m^2 - 1) k^2 ]

This can be factored as:

[ 2(m k)(m - k) 4 ]

Considering the factor pairs of 4, we have:

(1, 4) (2, 2) (-1, -4) (-2, -2)

Solving the resulting cases, we find that the only valid solution is [ m 1 ].

Substituting back, we get:

[ 2n^4 - 2n^2 - 1 1 ]

This simplifies to:

[ 2n^4 - 2n^2 0 ]

Factoring, we obtain:

[ 2n^2(n^2 - 1) 0 ]

This gives:

[ n 0 text{ or } n 1 ]

Checking for Valid Prime Numbers

If [ n 0 ], then:

[ p 2(0)^2 - 1 -1 ]

This is not a prime number.

If [ n 1 ], then:

[ p 2(1)^2 - 1 1 ]

This is also not a prime number.

Checking for higher values of [ n ], we find:

For ( n 2 ):

[ p 2(2)^2 - 1 7 quad text{and} quad frac{7^2 - 1}{2} frac{49 - 1}{2} 25 text{ is a perfect square.} ]

For ( n 3 ):

[ p 2(3)^2 - 1 17 quad text{and} quad frac{17^2 - 1}{2} frac{289 - 1}{2} 145 text{ is not a perfect square.} ]

Continuing this process, we find no other prime numbers that satisfy both conditions. After checking several values, the only prime number that satisfies both conditions is ( p 7 ).

Conclusion

The sum of all such primes ( p ) is:

7

This process showcases the intricate mathematical techniques used to solve complex problems and highlights the beauty of prime numbers and their properties.