P is a (positive) prime number such that 4p+1 is a perfect cube.what is the sum of all possible value of p?
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Given : P is a (positive) prime number such that 4p+1 is a perfect cube.
To find : sum of all possible value of p
Solution:
4p + 1 = a³
=> 4p = a³ - 1
using x³ - y³ = (x - y)(x² + xy + y²)
=> 4p = (a - 1)(a ² + a + 1)
a ² + a + 1 = a(a + 1) + 1 Hence always odd as a(a + 1) is always even
=> a - 1 = 4 as p is prime
=> a = 5
=> 4p + 1 = 5³
=> 4p = 124
=> p = 31
only possible value of p = 31
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