Australia
Let ^@ p ^@ be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers. ^@(^@For example, if you think ^@ 11 ^@ and ^@ 13 ^@ are two such prime numbers, then the sum is ^@ 24.) ^@
Answer:
^@ 3 ^@
- Given, ^@ p ^@ is a prime number such that the next larger number is a perfect square.
^@ i.e., p + 1 = n^2 ^@ - Rewriting ^@ p + 1 = n^2, ^@ we get
^@ \begin{align} & p = n^2 - 1 \\ \implies & p = (n - 1)(n +1) \end{align} ^@ - Since ^@ p ^@ is a prime number, therefore ^@ n-1 ^@ needs to be equal to ^@ 1. ^@
^@ \implies n = 2 ^@
Therefore, ^@ p = 3 ^@ is unique.
Hence, the required sum is ^@ 3. ^@
