Exploring Proof by Contradiction for Infinite Primes

This tutorial demonstrates a proof by contradiction to show that there are infinitely many prime numbers. It begins by assuming the opposite, that there is a finite number of primes, and lists them. A new number is constructed by multiplying all listed primes and adding one, which cannot be divided by any of the listed primes, leading to a contradiction. This proves the original statement that there are infinitely many primes. Key points include writing the negation of the statement and identifying the contradiction.

What is the first step in a proof by contradiction?

• Assume the statement is true

• Assume the opposite of the statement is true

• List all possible outcomes

• Find a counterexample

What do we assume about the number of prime numbers in this proof?

• There are no prime numbers

• There are a finite number of prime numbers

• Prime numbers are all even

• There are infinitely many prime numbers

What is the notation used for the list of prime numbers?

• N1, N2, N3, ..., NN

• X1, X2, X3, ..., XN

• A1, A2, A3, ..., AN

• P1, P2, P3, ..., PN

What operation is performed on the list of prime numbers?

• Addition

• Subtraction

• Multiplication

• Division

What is added to the product of all listed prime numbers?

• 3

• 1

• 0

• 2

What is the significance of the remainder when the new number is divided by any of the listed primes?

• It is always 2

• It is always 0

• It is always 3

• It is always 1

Why can't any of the listed primes be a factor of the new number?

• Because the new number is even

• Because the new number is odd

• Because the new number leaves a remainder of 1

• Because the new number is prime

What does the existence of another prime number not in the original list indicate?

• There are infinitely many prime numbers

• Prime numbers are all even

• There are finitely many prime numbers

• There are no prime numbers

What should be explicitly written down in a proof by contradiction?

• The original statement

• The negation of the statement

• The final conclusion

• The list of prime numbers

What is the final step in a proof by contradiction?

• Conclude that the original statement is true

• Find a counterexample

• Assume the statement is false

• List all possible outcomes