Finding primes given a condition

In summary, the question asks to find all prime numbers p such that p^2 = n^3 + 1 for some integer n. This can be solved by recognizing that p^2 can only be p times p or 1 times p^2. This leads to two possible solutions: one where one of the factors is equal to 1 and the other is equal to p^2, and another where both factors are equal to each other. Another approach is to recognize that all odd p^2 is usually of the form p^2 = 1 mod 4, and n^3 + 1 = 1 mod 4 or n^3 = 0 mod 4. This method requires a list of primes
  • #1
galois427
16
0
Hi, I need help solving this problem. The question asks me to find all prime numbers p such that p^2 = n^3 + 1 for some integer n.
 
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  • #2
This is really quite easy as p^2 can only be p times p or 1 times p^2.

Now, [tex]n^3 + 1 = (n + 1)(n^2 - n + 1)[/tex]

So one possible solution is where one of the factors is equal to 1 and the other is equal to p^2. Or when they are both equal to each other. Test those out and you should find all the possible solutions for n.
 
  • #3
An alternative,
all odd p^2 is usually of the form
p^2 = 1 mod 4

n^3+1 = 1 mod 4
or n^3 = 0 mod 4

Now this is useful if and only if u have a list of primes ...
So first u can generate a list and then check for conditions ...

-- AI
 

1. How do you define a prime number?

A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has no other factors besides 1 and itself.

2. What is the condition for finding primes?

The condition for finding primes is a criteria or rule that determines which numbers are considered prime and which are not. This can vary depending on the specific problem or scenario.

3. Can you give an example of a condition for finding primes?

Sure, one example of a condition for finding primes is the "twin prime conjecture", which states that there are infinitely many pairs of prime numbers that differ by 2, such as 41 and 43 or 71 and 73.

4. Is there a specific method or algorithm for finding primes given a condition?

Yes, there are several methods and algorithms for finding primes given a condition. Some common ones include the Sieve of Eratosthenes, the Fermat Primality Test, and the Miller-Rabin Primality Test.

5. Why is finding primes with a condition important in mathematics?

Finding primes with a condition can help mathematicians understand the patterns and properties of prime numbers, which are considered the building blocks of all positive integers. It can also have practical applications in fields such as cryptography and computer science.

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