Let N be the sum of all positive integers q of the form q=p^k with prime p, such that for at least four different integer values of x from 1 to q,
x^3−3x≡123(modq).
What are the last 3 digits of N?
To find the last three digits of N, we need to find the sum of all positive integers q of the given form that satisfy the congruence equation for at least four different integer values of x from 1 to q.
Let's break down the problem into smaller steps:
Step 1: Find the prime numbers p that satisfy the congruence equation for at least four different integer values of x from 1 to q.
We know that the equation x^3 - 3x ≡ 123 (mod q) can be rewritten as x^3 - 3x - 123 ≡ 0 (mod q). So, we are looking for prime numbers p such that x^3 - 3x - 123 ≡ 0 (mod p^k) has at least four solutions for x.
Step 2: For each prime p, find the smallest positive integer k such that the equation x^3 - 3x - 123 ≡ 0 (mod p^k) has at least four solutions for x. Let's call this k_p.
To find k_p, we can start with k = 1 and increase it until we find at least four solutions. To find the solutions, we can try all possible values of x from 1 to p^k.
Step 3: For each prime p, calculate p^k_p and add it to the sum N.
Step 4: Finally, calculate the last three digits of N by taking N modulo 1000.
This involves a systematic approach of checking prime numbers, finding k_p values, and calculating the sum N. It is a complex task that requires multiple iterations and calculations.