Consider the following incorrect theorem: Suppose n is a...

Consider the following incorrect theorem:
Suppose $n$ is a natural number larger than 2, and $n$ is not a prime number. Then $2 n+13$ is not a prime number.
What are the hypotheses and conclusion of this theorem? Show that the theorem is incorrect by finding a counterexample.

Key Concepts

Logical Structure of Theorems

Any mathematical theorem is made up of a set of premises (hypotheses) and a conclusion that logically follows if the premises are true. This structure is essential for understanding and verifying the validity of a mathematical argument.

Hypotheses

The hypotheses are the conditions or assumptions that are assumed to be true in order for the theorem to be applied. They set the context and limitations for the statement and are critical for deriving the conclusion.

Conclusion

The conclusion is the statement or assertion that is claimed to follow from the hypotheses. It is what one aims to prove based on the assumed conditions provided by the hypotheses.

Counterexample Method

A counterexample is a specific case that violates the conclusion of a theorem while satisfying its hypotheses. It is a powerful tool in disproving universal statements or incorrect theorems since finding even one counterexample is enough to demonstrate that the theorem does not hold in all cases.

Natural Numbers

Natural numbers are a fundamental set of numbers typically used for counting and ordering. In many mathematical contexts, including number theory, properties of natural numbers are central to the formulation and analysis of problems.

Prime Numbers

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. They play a crucial role in number theory due to their fundamental building block status in the factorization of integers.

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