We return to the rule we discovered: The Unique Prime Factorisation: If we the order of the factors, don’t mind Any positive natural number can be written in a unique way as a product of primes with indices. Class survey/discussion: a) Think about the time when we drew different factor trees for 48 and we always found the same with same indices. Did this surprise you? “hanging fruits” If yes, why? If not, why not? b) Think about the Unique Prime Factorisation rule. Are you convinced that it’s true for all natural numbers? If yes, why? If not, why not? c) As the name suggests, the Unique Prime Factorisation is due to the very special nature of prime numbers. reformulate this rule in terms of primes: Let’s Suppose that we write a number as a product , and start a factor tree like this: Moderator’s Comments: a) Some people may not have been surprised by this discovery. This may be because they already knew all the factors of 48 and could quickly check that they have no prime factors other than 2 and 3. Or they may have noticed that 48 could not possibly have any prime factors other than 2 and 3: indeed, the only prime they needed to check would be 5, since 7 already satisfies These checks can go on so quickly in our minds that we may find the results absolutely natural and obvious... b) Even though a rule holds for 48 and other numbers small enough to check, this does not guarantee that the rule will be true for all numbers. c) Conclusions: The following scary questions follow directly from our discussion: d) Is Unique Prime Factorisation true even for very large numbers, with very large prime factors? e) How large can prime numbers get anyway? Are there infinitely many primes? f) We reformulated the Unique Prime Factorisation rule as a property of primes:
We return to the rule we discovered: The Unique Prime Factorisation: If we don’t mind the order of the factors, Any positive natural number can be written in a unique way as a product of primes with indices. Class survey/discussion: a) Think about the time when we drew different factor trees for 48 and we always found the same “hanging fruits” with same indices. Did this surprise you? If yes, why? If not, why not? b) Think about the Unique Prime Factorisation rule. Are you convinced that it’s true for all natural numbers? If yes, why? If not, why not? c) As the name suggests, the Unique Prime Factorisation is due to the very special nature of prime numbers. Let’s reformulate this rule in terms of primes: Suppose that we write a number as a product , and start a factor tree like this: Moderator’s Comments: a) Some people may not have been surprised by this discovery. This may be because they already knew all the factors of 48 and could quickly check that they have no prime factors other than 2 and 3. Or they may have noticed that 48 could not possibly have any prime factors other than 2 and 3: indeed, the only prime they needed to check would be 5, since 7 already satisfies These checks can go on so quickly in our minds that we may find the results absolutely natural and obvious... b) Even though a rule holds for 48 and other numbers small enough to check, this does not guarantee that the rule will be true for all numbers. c) Conclusions: The following scary questions follow directly from our discussion: d) Is Unique Prime Factorisation true even for very large numbers, with very large prime factors? e) How large can prime numbers get anyway? Are there infinitely many primes? f) We reformulated the Unique Prime Factorisation rule as a property of primes: