Here is a theorem about prime numbers: \"Let n be an integer. If 2^n - 1 is prim

Question

Here is a theorem about prime numbers: "Let n be an integer. If 2^n - 1 is prime, then n is prime, then n is prime" Compute 2^11 - 1 and show that it is not prime. Explain why part i. does not contradict the theorem. The scene is a courtroom. The prosecutor says "If the defendant is guilty, then he had an accomplice.". The defence attorney leaps up and exclaims "That's not true!". The judge accepts that the defence attorney's statement is true and sentences the defendant to a year in jail. Explain the judge's reasoning. Three logicians walk into a bar. The bartender asks "Does everyone want a beer?" The first logician says "I don't know." So does the second logician. The third logician then says "Yes!". Is he right? Explain.

Explanation / Answer

Theorem --- if 2^n-1 is prime then n is prime .

2^11-1 = 2047=23*89

Which implies that 2^11-1 is not prime

(ii) theorm says if 2^n-1 is prime then n is prime

And this is the counter example for the converse part of the theorem 11 is prime but 2^11-1 is not .this doesn't contradict the given theorem.

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