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Old 2019-11-23, 19:19   #325
sweety439
 
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Originally Posted by sweety439 View Post
Not including the single-digit primes, proof of that these sets are complete:

b=2: we obtain the 2-digit primes 10 and 11, since for any prime p > 11, p must start and end with digit 1, and we have 11 <<< p, thus the 2-kernel {10, 11} is complete.

b=3: we obtain the 2-digit primes 10, 12 and 21, for any prime p > 21, if p end with 2 and 12 !<<< p, then p must contain only 0 and 2, thus p is divisible by 2 and > 2, thus not prime, therefore, p must end with 1 (p cannot end with 0, or p is divisible by 10 and not prime), since 21 !<<< p, p must contain only 0 and 1, but since 10 !<<< p and p cannot have leading zeros, thus p can only have the digit 1, i.e. p is a repunit, and the smallest repunit prime is 111, thus completed the 3-kernel {10, 12, 21, 111}.

b=4: we obtain the 2-digit primes 11, 13, 23 and 31, for any prime p > 31, if p end with 3 and 13 !<<< p and 23 !<<< p, then p must contain only 0 and 3, thus p is divisible by 3 and > 3, thus not prime, therefore, p must end with 1 (p cannot end with 0 or 2, or p is divisible by 2 and not prime), since 11 !<<< p and 31 !<<< p, thus p (before the final digit 1) must contain only 0 and 2, and we obtain the prime 221, since p cannot have leading zeros, the remain case is only 2{0}1, but all numbers of the form 2{0}1 are divisible by 3 and > 3, thus not prime, thus we completed the 4-kernel {11, 13, 23, 31, 221}.

b=5: we obtain the 2-digit primes 10, 12, 21, 23, 32, 34 and 43, for any prime p>43:

p end with 1 --> before this 1, p cannot contain 2 --> if p end with 11, then we find the prime 111, and all other primes contain at most two 1, ...

p end with 2 --> before this 2, p cannot contain 1 or 3 --> p only contain 0, 2 and 4 --> p is divisible by 2 and > 2 --> p is not prime (thus, 12 and 32 are the only such primes end with 2)

p end with 3 --> before this 3, p cannot contain 2 or 4 --> p only contain 0, 1 and 3 --> we obtain the primes 133 and 313, thus other primes p cannot contain both 1 and 3 (before the final digit 3), and since p cannot have leading zeros, if p begin with 1, then p is of the form 1{0,1}3 --> it must be of the form {1}3 (to avoid the prime 10) --> but 113 is not prime and all primes except 111 contain at most two 1 --> this way cannot find any primes, if p begin with 3, then p is of the form 3{0,3}3 --> p is divisible by 3 and > 3 --> p is not prime (thus, all such primes end with 3 are 23, 43, 133 and 313)

p end with 4 --> before this 4, p cannot contain 3 --> since all primes > 2 are odd, p must contain at least one 1 --> we obtain the prime 414 and we know that no 2 can before this 1 (to avoid the prime 21) and no 0 or 2 can after this 1 (to avoid the primes 10 and 12) --> 1 must be the leading digit (since p cannot have leading zeros, and no 2, 3, 4 can before this 1 (to avoid the primes 21, 34 and 414, respectively) --> p must be of the form 1{4} or 11{4} (since all primes except 111 contain at most two 1) --> and we obtain the prime 14444 (thus, all such primes end with 4 are 34, 414 and 14444)
b=5, p end with 1: (assume p is a prime in the minimal set of the strings for primes with at least two digits in base 5 other than 10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031)

* before this 1, p cannot contain 2 (because of 21)

* if before this 1, p contain 1 --> assume p is {xxx}1{yyy}1 --> y cannot contain 0, 1, 2, or 3 (because of 10, 111, 12, and 131)

** if y is not empty --> y contain only the digits 4 --> x cannot contain 1, 2, 3, or 4 (because of 111, 21, 34, and 414) --> x contain only the digits 0 (a contradiction, since a number cannot have leading zeros, and all numbers of the form 1{4}1 is divisible by 2 and cannot be primes)

** thus y is empty --> p is {xxx}11 --> x cannot contain 1 or 2 (because of 111 and 21), but x cannot contain both 3 and 4 (because of 34 and 43) --> x contain either only 0 and 3, or only 0 and 4, however, {0,3}11 is divisible by 3, and {0,4}11 is divisible by 2, and neither can be primes, a contradiction!!!

--> thus, before this 1, p cannot contain 1

Last fiddled with by sweety439 on 2019-11-24 at 15:23
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