Prove primes based on associates

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So, in summary, if a is a prime number, then b is also a prime number because they are associates and b can only be 1 or itself.
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Homework Statement



Suppose that a and b are associates. Prove that if a is a prime number then b is a prime number.

Homework Equations



Associates are two numbers that are exactly equal when the abs(a) = abs(b). Or, re-write as a=ub where u is a unit.

The Attempt at a Solution



Basically started with trying to use the fact that the absolute values of the two numbers had to be the same. If that is the case, then the numbers are related and therefore b is a prime.

This just seems like I'm missing a step or two. I don't have anything to really tie it together.
 
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  • #2
Let a = p, a prime number. Then, b = up for some unit u. Since p is prime, it can only be divided by 1 and itself, so p = 1 or p = p. Since b = up, then b = u1 or b = up. But, since u is a unit, then u = 1 or u = -1. Therefore, if a is a prime number, then b must also be a prime number because b = 1 or b = p.
 

What is the concept of proving primes based on associates?

The concept of proving primes based on associates is a mathematical technique used to show that a number is a prime number by demonstrating that it has no other divisors except for 1 and itself. This is achieved by showing that the number's associates, which are other numbers that are generated by rearranging the digits of the original number, also have no divisors.

How is this concept different from other methods of proving primes?

This concept differs from other methods of proving primes, such as the Sieve of Eratosthenes or the Fermat's Little Theorem, because it focuses on the relationship between a number and its associates. It relies on the fact that if a number is prime, then all of its associates will also be prime. This makes it a more efficient method for proving primes for larger numbers.

What are the advantages of using this method?

One of the main advantages of using this method is that it can be applied to larger numbers compared to other methods. It also provides a more direct proof of a number's primality, as it relies on the properties of the number itself rather than any external factors. Additionally, it allows for a more intuitive understanding of a number's primality by examining its associates.

What are some examples of proving primes based on associates?

An example of proving primes based on associates is the number 17. Its associates are 71 and 107, and since all three numbers are prime, we can conclude that 17 is also a prime number. Another example is the number 113, whose associates are 131 and 311. Since all three numbers are prime, we can prove that 113 is also a prime number.

Are there any limitations to this method?

One potential limitation of this method is that it requires a certain level of mathematical knowledge and understanding to apply it effectively. It also may not be applicable to all numbers, as some numbers may not have any associates that are also prime. Lastly, it may not be the most efficient method for proving primes for smaller numbers, as other methods may be more suitable in those cases.

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