A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to $1$.
Define $\mathcal{F}$ to be the set of all monic polynomials with integer coefficients (including the constant polynomial $p(x)=1$). A polynomial $p(x)\in\mathcal{F}$ is cyclogenic if there exists $q(x)\in\mathcal{F}$ and a positive integer $n$ such that $p(x)q(x)=x^n-1$. If $n$ is the smallest such positive integer then $p(x)$ is $n$-cyclogenic.
Define $P_n(x)$ to be the sum of all $n$-cyclogenic polynomials. For example, there exist ten 6-cyclogenic polynomials (which divide $x^6-1$ and no smaller $x^k-1$):
$$\begin{align*}
&x^6-1&&x^4+x^3-x-1&&x^3+2x^2+2x+1&&x^2-x+1\\
&x^5+x^4+x^3+x^2+x+1&&x^4-x^3+x-1&&x^3-2x^2+2x-1\\
&x^5-x^4+x^3-x^2+x-1&&x^4+x^2+1&&x^3+1\end{align*}$$
giving
$$P_6(x)=x^6+2x^5+3x^4+5x^3+2x^2+5x$$
Also define
$$Q_N(x)=\sum_{n=1}^N P_n(x)$$
It’s given that
$Q_{10}(x)=x^{10}+3x^9+3x^8+7x^7+8x^6+14x^5+11x^4+18x^3+12x^2+23x$ and $Q_{10}(2) = 5598$.
Find $Q_{10^7}(2)$. Give your answer modulo $1\,000\,000\,007$.
To solve this problem, we need to implement a few functions in Python. We will define four functions to solve this problem:
1. `is_monic`: It checks if a polynomial is monic or not.
2. `is_cyclogenic`: It checks if a polynomial is cyclogenic or not.
3. `get_cyclogenic_polynomials`: It returns a list of all cyclogenic polynomials for a given positive integer `n`.
4. `calculate_Qn`: It calculates the polynomial Qn(x) for a given positive integer `n`.
Let’s start by implementing the first function, `is_monic`:
“`python
def is_monic(poly):
return poly[-1] == 1
“`
The function checks if the coefficient of the highest degree term in the polynomial `poly` is equal to `1`. If it is, then the polynomial is monic.
Next, let’s implement the `is_cyclogenic` function:
“`python
def is_cyclogenic(p, n):
for q in get_cyclogenic_polynomials(n):
if p * q == [0] + [1] + [0] * (n – 1):
return True
return False
“`
This function takes a polynomial `p` and a positive integer `n` as input. It checks if there exists a polynomial `q` such that `p * q` is equal to the polynomial `[0] + [1] + [0] * (n – 1)`. If such a polynomial exists, then `p` is cyclogenic.
Now, let’s implement the `get_cyclogenic_polynomials` function:
“`python
def get_cyclogenic_polynomials(n):
cyclogenic_polynomials = []
for d in range(1, n + 1):
if n % d == 0:
poly = [0] * (n // d) + [-1] + [0] * (d – n // d – 1) + [1]
if is_monic(poly):
cyclogenic_polynomials.append(poly)
return cyclogenic_polynomials
“`
This function generates all the possible cyclogenic polynomials for a given positive integer `n`. It uses the fact that a cyclogenic polynomial has to divide `x^n – 1`. The function iterates over all the divisors of `n` and generates the corresponding polynomial using the pattern `[0] * (n // d) + [-1] + [0] * (d – n // d – 1) + [1]`. The generated polynomial is then checked if it is monic using the `is_monic` function. If it is monic, it is added to the list of cyclogenic polynomials.
Finally, let’s implement the `calculate_Qn` function:
“`python
def calculate_Qn(n):
Qn = [0] * (n + 1)
for i in range(1, n + 1):
cyclogenic_polynomials = get_cyclogenic_polynomials(i)
for poly in cyclogenic_polynomials:
Qn = [(Qn[j] + poly[j]) % 1000000007 for j in range(i + 1)]
return Qn
“`
This function calculates the polynomial Qn(x) for a given positive integer `n`. It iterates from `1` to `n` and for each `i`, it gets the list of cyclogenic polynomials for `i` using the `get_cyclogenic_polynomials` function. It then adds each cyclogenic polynomial to the corresponding degree term of Qn(x) by taking modulo 1000000007 to keep the calculations within the given constraints.
Now, we can calculate Q_{10^7}(2) by calling the `calculate_Qn` function:
“`python
result = calculate_Qn(10**7)[2]
“`
The result will be the coefficient of x^2 in Q_{10^7}(x) modulo 1000000007.
Note: Since the calculations involve large numbers, you may want to use libraries like numpy or sympy for efficient polynomial arithmetic and modulo calculations.
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