5 Charlie claims to have proved the following statement.
"The sum of a square number and a prime number cannot be a square number."
- Give an example to show that Charlie's statement is not true.
Charlie's attempt at a proof is below.
Assume that the statement is not true.
⇒ There exist integers \(n\) and \(m\) and a prime \(p\) such that \(n ^ { 2 } + p = m ^ { 2 }\).
\(\Rightarrow p = m ^ { 2 } - n ^ { 2 }\)
\(\Rightarrow p = ( m - n ) ( m + n )\)
\(\Rightarrow p\) is the product of two integers.
\(\Rightarrow p\) is not prime, which is a contradiction.
⇒ Charlie's statement is true. - Explain the error that Charlie has made.
- Given that 853 is a prime number, find the square number \(S\) such that \(S + 853\) is also a square number.