How to Prove a Congruence in Modulo pq with Distinct Primes?

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In summary, the prime congruence problem is a mathematical problem that involves finding solutions to a system of congruences with prime moduli. It is difficult due to the complexity of the equations and requires advanced techniques to solve. This problem has significant implications in number theory, cryptography, and coding theory. Currently, there is no known general solution, but researchers are continuously working to find new techniques and explore its connections to other areas of mathematics.
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Question: Suppose p and q are distinct primes. Show that p^(q-1) + q^(p-1) is congruent to 1 modulo pq.

Answer: I know from Little Fermat Theorem that p^(q-1) is congruent to 1 modulo q and q^(p-1) is congruent to 1 modulo p, but I have no idea how to combine these two.
 
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You know from the CRT that there is only one residue class mod pq that gives you A mod p and B mod q. So if 1 mod pq gives 1 mod p and 1 mod q, then you're done.
 

FAQ: How to Prove a Congruence in Modulo pq with Distinct Primes?

What is the prime congruence problem?

The prime congruence problem, also known as the prime residue problem, is a mathematical problem that involves finding solutions to a system of congruences in which all the moduli are prime numbers. It asks whether there exists a solution to the system of congruences and if so, how many solutions are there.

What makes the prime congruence problem difficult?

The prime congruence problem is difficult because it involves finding solutions to a system of equations with prime moduli, which can be very large and complex. It also requires advanced mathematical techniques and algorithms to solve, making it a challenging problem for mathematicians.

What is the importance of the prime congruence problem?

The prime congruence problem has significant implications in number theory, particularly in the study of prime numbers and their properties. It also has applications in cryptography and coding theory, making it an important problem in the field of mathematics.

Is there a known solution to the prime congruence problem?

No, there is currently no known general solution to the prime congruence problem. However, there are specific cases where solutions have been found, such as the Chinese Remainder Theorem which provides a solution for congruence systems with coprime moduli.

What are some current research efforts on the prime congruence problem?

There are ongoing research efforts to find new techniques and algorithms to solve the prime congruence problem, as well as to explore its connections to other areas of mathematics. Some researchers are also focused on finding specific solutions for certain types of congruence systems and exploring their applications in different fields.

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