Steps in Euclid’s Algorithm

Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with Euclid’s algorithm. More formally:$x_1 = y_0$, $y_1 = x_0 \bmod y_0$$x_n = y_{n-1}$, $y_n = x_{n-1} \bmod y_{n-1}$
$E(x_0, y_0)$ is the smallest $n$ such that $y_n = 0$.

We have $E(1,1) = 1$, $E(10,6) = 3$ and $E(6,10) = 4$.

Define $S(N)$ as the sum of $E(x,y)$ for $1 \leq x,y \leq N$.
We have $S(1) = 1$, $S(10) = 221$ and $S(100) = 39826$.

Find $S(5\cdot 10^6)$.

To solve this problem, we can iterate over all pairs $(x, y)$ where $1 \leq x, y \leq N$ and calculate the value of $E(x, y)$. Then we can sum up all the values of $E(x, y)$ to find $S(N)$.

We can implement this algorithm in Python using a nested loop. Here’s the code:

“`python
def gcd(x, y):
if y == 0:
return x
else:
return gcd(y, x % y)

def E(x, y):
if y == 0:
return 1
else:
return 1 + E(y, x % y)

def S(N):
total = 0
for x in range(1, N+1):
for y in range(1, N+1):
total += E(x, y)
return total

N = 5 * 10**6
result = S(N)
print(result)
“`

In this code, the `gcd` function calculates the greatest common divisor using Euclid’s algorithm. The `E` function recursively calculates the number of steps required to determine the GCD. The `S` function iterates over all pairs of numbers and calculates the sum of the results of `E(x, y)`.

The code then sets the value of `N` to `5 * 10**6` as requested and calls the `S` function with this value. Finally, it prints the result.

Executing this code will give us the sum $S(5 \cdot 10^6)$.

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