To solve the problem step by step, we will analyze each statement provided in the question.
### Step 1: Analyze Statement 1
We need to determine whether the integral part of \( (8 + 3\sqrt{7})^{20} \) is even.
1. **Define the terms**: Let \( x = 8 + 3\sqrt{7} \) and \( y = 8 - 3\sqrt{7} \).
2. **Calculate \( y \)**: Since \( \sqrt{7} \approx 2.64575 \), we have \( 3\sqrt{7} \approx 7.93725 \). Thus, \( y = 8 - 3\sqrt{7} \approx 0.06275 \).
3. **Evaluate \( y^{20} \)**: Since \( 0 < y < 1 \), \( y^{20} \) will be a very small positive number (between 0 and 1).
4. **Use the Binomial Theorem**: By the theorem, we can write:
\[
x^{20} + y^{20} = (8 + 3\sqrt{7})^{20} + (8 - 3\sqrt{7})^{20}
\]
The sum will yield an integer because it consists of the sum of even powers of \( (8 + 3\sqrt{7}) \) and \( (8 - 3\sqrt{7}) \).
5. **Determine the integral part**: Let \( I = (8 + 3\sqrt{7})^{20} \) and \( F = (8 - 3\sqrt{7})^{20} \). Thus, the integral part of \( I \) is given by:
\[
\lfloor I \rfloor = I + F - F
\]
Since \( F \) is very small, \( \lfloor I \rfloor = I + F - F \) is approximately \( I \) (which is an integer).
6. **Conclusion**: Since \( F \) is positive and less than 1, the integral part \( \lfloor I \rfloor \) is odd.
**Result**: Statement 1 is **False**.
### Step 2: Analyze Statement 2
We need to check if the sum of the last eight coefficients in the expansion of \( (1+x)^{16} \) is \( 2^{15} \).
1. **Identify the coefficients**: The coefficients of \( (1+x)^{16} \) are given by \( \binom{16}{k} \) for \( k = 0, 1, 2, \ldots, 16 \).
2. **Last eight coefficients**: The last eight coefficients are \( \binom{16}{8}, \binom{16}{9}, \ldots, \binom{16}{15}, \binom{16}{16} \).
3. **Use symmetry**: By the symmetry of binomial coefficients, we have:
\[
\binom{16}{k} = \binom{16}{16-k}
\]
Therefore, the last eight coefficients can be paired with the first eight coefficients:
\[
\binom{16}{0}, \binom{16}{1}, \ldots, \binom{16}{7}
\]
4. **Sum of all coefficients**: The sum of all coefficients in \( (1+x)^{16} \) is \( 2^{16} \).
5. **Calculate the sum of the first eight coefficients**: The sum of the first eight coefficients is equal to the sum of the last eight coefficients, so:
\[
2 \times \text{(sum of last 8 coefficients)} + \binom{16}{8} = 2^{16}
\]
6. **Final calculation**: Thus, the sum of the last eight coefficients is:
\[
\text{Sum} = \frac{2^{16} - \binom{16}{8}}{2}
\]
Since \( \binom{16}{8} = 12870 \), we find that the sum does not equal \( 2^{15} \).
**Result**: Statement 2 is **False**.
### Step 3: Analyze Statement 3
We need to determine if \( RF = 2^{2n+1} \) holds true.
1. **Define \( R \)**: Let \( R = (5\sqrt{5} + 11)^{2n+1} \).
2. **Greatest integer function**: We can express \( R \) as \( R = [R] + F \), where \( [R] \) is the greatest integer part and \( F \) is the fractional part.
3. **Evaluate \( F \)**: The fractional part \( F \) is given by \( R - [R] \).
4. **Calculate \( RF \)**: We can express:
\[
RF = (5\sqrt{5} + 11)^{2n+1} \cdot F
\]
5. **Use the difference of squares**:
\[
RF = \left( (5\sqrt{5} + 11)^{2n+1} \right) \left( (5\sqrt{5} - 11)^{2n+1} \right) = (125 - 121)^{2n+1} = 4^{2n+1} = 2^{2n+2}
\]
6. **Conclusion**: This indicates that \( RF \) is indeed \( 2^{2n+1} \).
**Result**: Statement 3 is **True**.
### Final Conclusion
- Statement 1: False
- Statement 2: False
- Statement 3: True
