Prime numbers proof by contradiction

In summary, the conversation discusses the proof by contradiction for the statement that for prime numbers $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. The conversation delves into the understanding of prime numbers being the product of two numbers that are either greater than one or less than the prime numbers. The conversation then presents the approach of considering the possible values of $c-b$ and $c+b$ for a prime $a$, leading to a contradiction.
  • #1
tmt1
234
0
For prime numbers, $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. Prove this by contradiction.

So, I get that $a^2 = c^2 - b^2 = (c - b)(c +b)$

And I get that prime numbers are the product of 2 numbers that are either greater than one, or less than the prime numbers.

But I'm unsure how to go from here.
 
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  • #2
tmt said:
For prime numbers, $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. Prove this by contradiction.

So, I get that $a^2 = c^2 - b^2 = (c - b)(c +b)$

And I get that prime numbers are the product of 2 numbers that are either greater than one, or less than the prime numbers.

But I'm unsure how to go from here.

a is prime so either $c-b = 1$ and $c+b= a^2$ or $c-b=c+b=a$ can you proceed from here
 

FAQ: Prime numbers proof by contradiction

What is a prime number?

A prime number is a positive integer that is divisible only by 1 and itself. In other words, it has exactly two distinct factors.

What is a proof by contradiction?

A proof by contradiction is a mathematical proof technique where we assume the opposite of what we want to prove and then show that it leads to a contradiction. This allows us to conclude that our original assumption must be true.

How does proof by contradiction apply to prime numbers?

In the proof by contradiction for prime numbers, we assume that there is a finite number of prime numbers and then show that this leads to a contradiction. This proves that there must be an infinite number of prime numbers.

What is the significance of proving the infinitude of prime numbers?

Proving the infinitude of prime numbers is significant because it is a fundamental concept in number theory and has many applications in other areas of mathematics and science. It also helps us understand the distribution of prime numbers and their role in the world of mathematics.

Are there any other methods of proving the infinitude of prime numbers?

Yes, there are other methods of proving the infinitude of prime numbers, such as Euclid's proof and Euler's proof. However, the proof by contradiction is one of the most commonly used and simplest methods to understand.

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