To determine the sequence of $\binom{n}{k} \mod 2$ for $k = 0 \ldots 7$ with $n = 1152921504606846987$, we need to use Lucas' theorem. Lucas' theorem gives a way to compute binomial coefficients modulo a prime number (in this case, 2) by looking at the binary representations of $n$ and $k$.

First, we convert $n$ to its binary representation: $n = 1152921504606846987$ $n \text{ in binary} = 100000000000000000000000000000000000000000000000000011$

We then compute the binomial coefficients $\binom{n}{k} \mod 2$ for $k = 0$ to $k = 7$ by examining the bits of $n$ and $k$.

Here are the binary representations:

• $k = 0$: $000$
• $k = 1$: $001$
• $k = 2$: $010$
• $k = 3$: $011$
• $k = 4$: $100$
• $k = 5$: $101$
• $k = 6$: $110$
• $k = 7$: $111$

Using Lucas' theorem, $\binom{n}{k} \mod 2$ is determined by the individual bits of $n$ and $k$. For each bit position where $k$ has a 1, $n$ must also have a 1 for $\binom{n}{k}$ to be odd (1 mod 2). If any bit position where $k$ has a 1 but $n$ has a 0, $\binom{n}{k}$ is even (0 mod 2).

Let's evaluate:

• For $k = 0$: always 1 (since $\binom{n}{0} = 1$).
• For $k = 1$: The rightmost bit of $n$ is 1.
• For $k = 2$: The second bit from the right of $n$ is 1.
• For $k = 3$: Both the first and second bits from the right of $n$ are 1.
• For $k = 4$: The third bit from the right of $n$ is 0.
• For $k = 5$: The third bit from the right of $n$ is 0.
• For $k = 6$: The third bit from the right of $n$ is 0.
• For $k = 7$: The third bit from the right of $n$ is 0.

Thus, we get the sequence: $\binom{n}{k} \mod 2$ for $k = 0$ to $k = 7$ as: $1, 1, 1, 1, 0, 0, 0, 0$

This can be written as the binary string: $11110000$

If you have any questions or need further details, feel free to ask!

Here are some related questions you might be interested in:

1. How does Lucas' theorem work for prime moduli other than 2?
2. Can you show a detailed example of Lucas' theorem for a smaller number?
3. How can we compute $\binom{n}{k} \mod 2$ for larger ranges of $k$?
4. What are the properties of binomial coefficients modulo a prime number?
5. How can Lucas' theorem be extended to composite moduli?

Tip: Lucas' theorem is a powerful tool for computing binomial coefficients modulo prime numbers, especially useful in combinatorial number theory and computer science for solving problems efficiently.