For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a
This is a tricky problem that combines number theory and combinatorics.
A function $f_{n,a,b}$ is called a retraction if $f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n$ for every $0 \le x < n$.
This is equivalent to checking the congruence $a(ax+b+nk)+b \equiv ax+b \mod n$ for $0 \le k < n$, which simplifies to $aa^2x+ab+a^2kn+ab \equiv ax+b \mod n$ for all $0 \le x < n$. Looking at the coefficients of $x$ on both sides, we see that $n | a^3 - a \implies n | a(a - 1)(a + 1)$. Since $0 < a < n$, the only possible solutions are $a = 1$. Therefore, the function $f_{n,a,b}$ is a retraction if and only if $a = 1$.
Each such function is characterized by its constant $b$. It can be checked easily that each function is a retraction. Therefore, for each $n > 1$, $R(n) = n$ because each $b$ from $0$ to $n – 1$ represents a different retraction.
In this case, we are asked to find $F(N) = \sum_{n = 1}^{N}(R(n^4 + 4))$. Since we’ve shown $R(n) = n$ for $n > 1$, $F(N) = \sum_{n = 1}^{N}(n^4 + 4) = \frac{N(N + 1)(2N + 1)(3N^2 + 3N – 1)}{30} + 4N$.
Given $F(1024) = 77532377300600$, we are asked to find $F(10^7)$. Substituting $1024$ and $10^7$ into that formula, the problem is reduced to performing large arithmetic computations and taking the result modulo $1\,000\,000\,007$.
To solve this problem, we need to calculate $F(10^7)$ and compute the result modulo $1\,000\,000\,007$. It leads to a large number, and depending on the limitations of your calculator or computation tool, it may go beyond what can be comfortably handled without specialized mathematical software.
Remember the following rules:
$$(a + b) \mod m = ((a \mod m) + (b \mod m)) \mod m$$
$$(a \times b) \mod m = ((a \mod m) \times (b \mod m)) \mod m$$
$$(a – b) \mod m = ((a \mod m) – (b \mod m)) \mod m$$
This properties allow us to calculate the modulo of a large number without having to explicitly evaluate it.
Hence, this problem would require writing a program (Python, for instance) to calculate $F(N)$ with large numbers and carry out the modulo operation. It requires an understanding of the mathematics behind the problem, knowledge of programming and handling large numbers in programming.
More Answers:
GCD and Tiling
The Inverse Summation of Coprime Couples
Eleven-free Integers
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This is a tricky problem that combines number theory and combinatorics.
A function $f_{n,a,b}$ is called a retraction if $f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n$ for every $0 \le x < n$. This is equivalent to checking the congruence $a(ax+b+nk)+b \equiv ax+b \mod n$ for $0 \le k < n$, which simplifies to $aa^2x+ab+a^2kn+ab \equiv ax+b \mod n$ for all $0 \le x < n$. Looking at the coefficients of $x$ on both sides, we see that $n | a^3 - a \implies n | a(a - 1)(a + 1)$. Since $0 < a < n$, the only possible solutions are $a = 1$. Therefore, the function $f_{n,a,b}$ is a retraction if and only if $a = 1$. Each such function is characterized by its constant $b$. It can be checked easily that each function is a retraction. Therefore, for each $n > 1$, $R(n) = n$ because each $b$ from $0$ to $n – 1$ represents a different retraction.
In this case, we are asked to find $F(N) = \sum_{n = 1}^{N}(R(n^4 + 4))$. Since we’ve shown $R(n) = n$ for $n > 1$, $F(N) = \sum_{n = 1}^{N}(n^4 + 4) = \frac{N(N + 1)(2N + 1)(3N^2 + 3N – 1)}{30} + 4N$.
Given $F(1024) = 77532377300600$, we are asked to find $F(10^7)$. Substituting $1024$ and $10^7$ into that formula, the problem is reduced to performing large arithmetic computations and taking the result modulo $1\,000\,000\,007$.
To solve this problem, we need to calculate $F(10^7)$ and compute the result modulo $1\,000\,000\,007$. It leads to a large number, and depending on the limitations of your calculator or computation tool, it may go beyond what can be comfortably handled without specialized mathematical software.
Remember the following rules:
$$(a + b) \mod m = ((a \mod m) + (b \mod m)) \mod m$$
$$(a \times b) \mod m = ((a \mod m) \times (b \mod m)) \mod m$$
$$(a – b) \mod m = ((a \mod m) – (b \mod m)) \mod m$$
This properties allow us to calculate the modulo of a large number without having to explicitly evaluate it.
Hence, this problem would require writing a program (Python, for instance) to calculate $F(N)$ with large numbers and carry out the modulo operation. It requires an understanding of the mathematics behind the problem, knowledge of programming and handling large numbers in programming.
More Answers:
GCD and TilingThe Inverse Summation of Coprime Couples
Eleven-free Integers