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First a definition: given a natural number ##a_na_{n-1}...a_0##, a subnumber is any number of the form ##a_k a_{k-1}...a_{l+1}a_l## for some ##0\leq l \leq k \leq n##. I think an example will be the easiest way to illustrate this definition: the subnumbers of ##1234## are

[tex]1,~2,~3,~4,~12,~23,~34,~123,~234,~1234[/tex]

Question 1: What is the largest possible number such that all subnumbers are prime? Is there such a largest possible number?

Some remarks: ##0## and ##1## are not prime. We work in the decimal system.

Question 2: How many numbers are there such that all subnumbers are prime.

Question 3: From now on we change the definitions accepting ##1## to also be a prime. What is now the largest number such that all subnumbers are prime? Is there a largest possible number now?

Question 4: How many numbers are there now?

Note: why allow ##1## to be prime? We have defined ##1## to be nonprime for good reasons, but there would also be good reasons to allow ##1## to be prime.

[tex]1,~2,~3,~4,~12,~23,~34,~123,~234,~1234[/tex]

Question 1: What is the largest possible number such that all subnumbers are prime? Is there such a largest possible number?

Some remarks: ##0## and ##1## are not prime. We work in the decimal system.

Question 2: How many numbers are there such that all subnumbers are prime.

Question 3: From now on we change the definitions accepting ##1## to also be a prime. What is now the largest number such that all subnumbers are prime? Is there a largest possible number now?

Question 4: How many numbers are there now?

Note: why allow ##1## to be prime? We have defined ##1## to be nonprime for good reasons, but there would also be good reasons to allow ##1## to be prime.

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