Project Euler Problem 66 Statement

Consider quadratic Diophantine equations of the form: $$x^2 - Dy^2 = 1$$

For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.

It can be assumed that there are no solutions in positive integers when $D$ is square.

By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we obtain the following:

Hence, by considering minimal solutions in $x$ for $D \le 7$, the largest $x$ is obtained when $D=5$.

Find the value of $D \le 1000$ in minimal solutions of $x$ for which the largest value of $x$ is obtained.

Solution

This script effectively solves the HackerRank version of Project Euler Problem 66, which involves calculating the largest minimal solution $x$ for Pell's equation $x^2 - D y^2 = 1$ across all $D \leq L$. Here's how the code addresses the problem step by step:

Key Components Breakdown:

1. Prime Sieve Optimization:
   • The prime_sieve function generates all prime numbers up to $L$. This step limits the computation of Pell's equation to prime $D$, avoiding unnecessary calculations for composite or square $D$, where trivial solutions exist.
   • The sieve operates by marking non-prime indices, reducing the time complexity to $O(n \log \log n)$.
2. Pell's Equation Solver:
   • The pell function computes the minimal solution $x$ for the given $D$. It applies a continued fraction expansion technique, iterating until the smallest non-trivial solution is found.
   • The use of integer arithmetic (//) ensures no floating-point precision errors, which is crucial for HackerRank's constraints and large $D$ values.
3. Iterating Over Primes:
   • The main logic takes the upper bound $L$ as input and iterates over all prime $D$ values returned by the sieve.
   • For each $D$, the script calculates the corresponding $x$ and keeps track of the maximum $x$ value and the corresponding $D$ using Python's max() function.
4. Efficient Output:
   • The script outputs the $D$ that produces the largest minimal solution $x$, which is the answer HackerRank requires.

Understanding Pell's Equation

Pell's equation $x^2 - D \, y^2 = 1$ is a classic Diophantine equation where $D$ is a positive integer that is not a perfect square. For each valid $D$, there is a fundamental solution $(x_1, y_1)$ with the smallest positive $x_1$ and $y_1$. Once that fundamental solution is known, all other solutions can be generated from it. In this particular problem, we must find among all $D$ up to the given limit $N$, which $D$ has the fundamental solution $x_1$ with the largest value.

The pell(d) Function

Inside the code, the pell(d) function implements a well-known algorithmic approach to solve for the fundamental solution $(x_1, y_1)$ in Pell's equation. It uses an adaptation of continued fractions (or a related process) to iteratively find the smallest $x_1$ and $y_1$ that satisfy $x^2 - d \, y^2 = 1$.

Variables:

• p and k track interim values during the continued fraction update.
• x1 and y track the current estimates for the Pell solution $(x, y)$.
• sd is the integer square root of $d$ (i.e., math.isqrt(d)).

Key Steps:

1. While k != 1 or y == 0, update p in a manner similar to the continued fraction expansion of $sqrt(d)$.
2. Recompute x1 and y using the updated p, thereby edging closer to the fundamental solution.
3. Update k using the relation $k = (p^2 - d)/k$.

Once the loop terminates, x1 holds the $x$-value of the fundamental solution of $x^2 - d \, y^2 = 1$. This iterative method circumvents having to build the entire continued fraction expansion explicitly and arrives at the result more directly.

Aggregating Results for All $d$

After defining pell(d), the script reads a single integer $N$ from input. It then calls Euler.prime_sieve(N), which produces all prime numbers up to $N$. For each prime $d$ in that range, the code computes (pell(d), d) and then finds the pair whose Pell solution pell(d) is largest.

Finally, it prints out only the $d$ (the second element in the pair) that produces the maximum x1 in the Pell equation solution. This meets the requirement of finding which $D &leq N$ yields the largest minimal $x$.

HackerRank version

Solves all test cases for Hackerrank Project Euler Problem 66

HackerRank Project Euler 66: Find the value of $D\lt N$, $(7\le N\le 10^4)$ in minimal solutions of $x$ for which the largest value of $x$ is obtained.

Last Word