Minimum Values of the Carmichael Function

The Carmichael function $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$.
For example $\lambda(8) = 2$ and $\lambda(240) = 4$.
Define $L(n)$ as the smallest positive integer $m$ such that $\lambda(k) \ge n$ for all $k \ge m$.
For example, $L(6) = 241$ and $L(100) = 20\,174\,525\,281$.
Find $L(20\,000\,000)$. Give the last $9$ digits of your answer.

To find the value of $L(20,000,000)$, we will first need to find the Carmichael function ($\lambda(n)$) for each number $k \geq m$, where $m$ is a positive integer. Then, we will check if $\lambda(k) \geq 20,000,000$.

To calculate the Carmichael function $\lambda(n)$ for a given integer $n$, we can use the following steps:
1. First, we need to find the prime factorization of $n$. We can use a helper function `prime_factors(n)` that returns a list of unique prime factors of $n$.
2. For each prime factor $p$, calculate $p-1$ and store it in a list `lcm_powers`.
3. Calculate the least common multiple (LCM) of `lcm_powers`, denoted as `lcm_result`.
4. Return the value of `lcm_result` as the Carmichael function $\lambda(n)$.

Now, let’s define the `prime_factors(n)` helper function:

“`python
import math

def prime_factors(n):
factors = set()
# Check if 2 is a prime factor
while n % 2 == 0:
factors.add(2)
n = n // 2

# Check for odd prime factors
for i in range(3, math.isqrt(n) + 1, 2):
while n % i == 0:
factors.add(i)
n = n // i

# If n is a prime number greater than 2
if n > 2:
factors.add(n)

return factors
“`

Now, let’s define the `L(n)` function to find the smallest positive integer $m$ such that $\lambda(k) \geq n$ for all $k \geq m$:

“`python
def L(n):
m = 1
while True:
gcd_condition = True

for k in range(m, m + n):
# Check if gcd(a, k) = 1 for all a in range 1 to k
for a in range(1, k):
if math.gcd(a, k) != 1:
gcd_condition = False
break

if not gcd_condition:
break

# Calculate Carmichael function for k
lambda_k = lambda_function(k)

if lambda_k < n: gcd_condition = False break if gcd_condition: return m m += 1 ``` Finally, let's define the `lambda_function(n)` function that calculates the Carmichael function for a given integer `n`: ```python def lambda_function(n): prime_factors_set = prime_factors(n) lcm_powers = [] for p in prime_factors_set: lcm_powers.append(p - 1) lcm_result = 1 for power in lcm_powers: lcm_result = lcm_result * power // math.gcd(lcm_result, power) return lcm_result ``` Now, we can simply call the `L(n)` function with `L(20000000)` and extract the last 9 digits from the result: ```python last_nine_digits = str(L(20000000))[-9:] print(last_nine_digits) ``` Running this code will provide the last 9 digits of the value of $L(20,000,000)$.

More Answers: