Proof using primes, divisibility, and sum of squares

In summary, the conversation is about proving or disproving two statements related to prime numbers and their divisibility with certain expressions. The first statement is about the relationship between a prime number and the difference of two squares, while the second statement is about the relationship between a prime number and the sum of two squares. The conversation also mentions looking for counterexamples as a helpful approach in solving the problem.
  • #1
ccsmarty
17
0

Homework Statement


I have to prove or disprove the following:

Part a) If p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 - c2)

Part b) f p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 + c2)

Homework Equations





The Attempt at a Solution



Part a)

Since p | (a2 + b2), we have that a2 + b2 = pk, for some integer k.
Since p | (c2 + d2), we have that c2 + d2 = pt, for some integer t.

Suppose p | (a2 - c2), then we have that a2 - c2 = pr, for some integer r.

By solving for a2 and c2 in the above equations, and substitution we have that
pk - b2 - (pt - d2) = pr
pk - pt - b2 + d2 = pr
pk - pt - pr = b2 - d2
p (k - t - r) = b2 - d2
So p | (b2 - d2)

I don't know where to go from here.

I figure that once I figure out how to do this part, the second part should be very similar.
Any help would be greatly appreciated. I'm going CRAZY trying to figure this out...
 
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  • #2
Did you try looking for counterexamples before going CRAZY? That's always a good idea.
 

1. What is the concept of using primes in proof?

The concept of using primes in proof involves utilizing the fundamental theorem of arithmetic, which states that every positive integer can be uniquely expressed as a product of primes. This allows for a more concise and efficient way of proving statements involving divisibility.

2. How does divisibility play a role in proof using primes?

Divisibility is essential in proof using primes because it allows us to break down a given number into its prime factors, making it easier to analyze and prove statements about the number. Divisibility also helps us identify common factors and patterns among numbers.

3. What is the significance of using the sum of squares in proof?

The sum of squares is a useful tool in proof because it allows us to identify certain properties and patterns of numbers. For example, the sum of two squares cannot be a prime number, except for specific cases such as 5.

4. Can proof using primes, divisibility, and sum of squares be applied to real-life situations?

Yes, the concepts of proof using primes, divisibility, and sum of squares are widely applicable in various fields such as cryptography, number theory, and computer science. It can also be used to solve real-life problems involving factors and divisibility.

5. Are there any limitations to using proof using primes, divisibility, and sum of squares?

While proof using primes, divisibility, and sum of squares is a powerful tool, there are certain limitations. For example, it may not be suitable for proving statements involving irrational numbers or complex equations. It also requires a solid understanding of number theory concepts, which can be challenging for some individuals.

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