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Number theory

Today at my number theory lecture the professor want to prove that the statement below is false.

Every prime number \(p\) can be written in the form \(p = ax+b\) were \(a, b\) are coprime integers, \(a > 1\) and \(x \in \mathbb N\).

She said that we need to find one case where the statement doesn't doesn't hold to be true.

The case she showed was \(4\cdot3+3=15 \) and since 15 is a composite then the statement is false.

I tried to explain to her that this proof is wrong but I couldn't.

So i want to mathematically explain why this proof is wrong, Any help with that?

Note by Mohamed Hassan
5 months ago

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Since 15 is a composite number, your professor's example of \(4 \cdot 3 + 3 = 15\) has no bearing on whether the given statement about primes is true or false.

Imagine I want to disprove the statement "All sheep are black." I point to a black swan, and say that since the swan is black, not all sheep are black. This argument is completely absurd, but it's the equivalent of what your professor is saying.

To disprove the given statement, I would have to find a certain prime \(p\), and show that it cannot be expressed in the form \(p = ax + b\).

Having said all this, the statement about primes seems odd to me. I would double-check to make sure that this is what your professor meant to say. Jon Haussmann · 4 months, 3 weeks ago

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