For which of the following integers $n$, there is at least one integer $x$ such that $x^2 \equiv -1 \pmod{n}$?

$ 
\textbf{(A)}\ 97
\qquad\textbf{(B)}\ 98
\qquad\textbf{(C)}\ 99
\qquad\textbf{(D)}\ 100
\qquad\textbf{(E)}\ \text{None of the preceding}
$
To determine for which of the given integers \( n \) there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{n} \), we need to analyze each option.

1. **Option (A): \( n = 97 \)**
   - \( 97 \) is a prime number.
   - We need to check if there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{97} \).
   - We know that for a prime \( p \equiv 1 \pmod{4} \), there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{p} \).
   - Since \( 97 \equiv 1 \pmod{4} \), there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{97} \).
   - For example, \( 22^2 = 484 \equiv -1 \pmod{97} \).

2. **Option (B): \( n = 98 \)**
   - \( 98 = 2 \times 7^2 \).
   - We need to check if there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{98} \).
   - Since \( 7 \equiv 3 \pmod{4} \), and for a prime \( p \equiv 3 \pmod{4} \), \( x^2 \neq -1 \pmod{p} \).
   - Therefore, \( x^2 \neq -1 \pmod{7} \), and hence \( x^2 \neq -1 \pmod{98} \).

3. **Option (C): \( n = 99 \)**
   - \( 99 = 3^2 \times 11 \).
   - We need to check if there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{99} \).
   - Since \( 11 \equiv 3 \pmod{4} \), and for a prime \( p \equiv 3 \pmod{4} \), \( x^2 \neq -1 \pmod{p} \).
   - Therefore, \( x^2 \neq -1 \pmod{11} \), and hence \( x^2 \neq -1 \pmod{99} \).

4. **Option (D): \( n = 100 \)**
   - \( 100 = 2^2 \times 5^2 \).
   - We need to check if there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{100} \).
   - Since \( x^2 \equiv -1 \pmod{4} \) is not possible (as the quadratic residues modulo 4 are 0 and 1), \( x^2 \neq -1 \pmod{100} \).

5. **Option (E): None of the preceding**
   - This option is incorrect because we have already found that \( x^2 \equiv -1 \pmod{97} \) is possible.

Conclusion:
The only integer \( n \) for which there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{n} \) is \( 97 \).

The final answer is \(\boxed{97}\).