Are numbers in which all substrings are primes finite regardless of base?
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For no particular reason I stumbled upon the sequence/set of all numbers where all substrings of the decimal representation is prime (A085823). It's quite easy to see that this must be a finite set. My approach was that the digits must alter between 3 and 7 eventually (since a prime can't end with 2 or 5 for larger than that and a prime can't consist of just two non-1 digits, those would be dividable by 2, 5 or 11) and we can see where it ends.
But if we change the base that approach fails. For example in base 8 a prime can end in 3, 5 or 7 AFAICS so such numbers can end in a altering sequence of 3, 5 and 7s but having tree digits to alter between there's always a way to vary the sequence.
I tried to write a python snippet to produce such numbers in different bases and for those I've tested the program seem to hang (ie don't seem to find any more numbers). Is it true that there are only a finite number of such numbers for every base?
number-theory prime-numbers
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
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add a comment |
$begingroup$
For no particular reason I stumbled upon the sequence/set of all numbers where all substrings of the decimal representation is prime (A085823). It's quite easy to see that this must be a finite set. My approach was that the digits must alter between 3 and 7 eventually (since a prime can't end with 2 or 5 for larger than that and a prime can't consist of just two non-1 digits, those would be dividable by 2, 5 or 11) and we can see where it ends.
But if we change the base that approach fails. For example in base 8 a prime can end in 3, 5 or 7 AFAICS so such numbers can end in a altering sequence of 3, 5 and 7s but having tree digits to alter between there's always a way to vary the sequence.
I tried to write a python snippet to produce such numbers in different bases and for those I've tested the program seem to hang (ie don't seem to find any more numbers). Is it true that there are only a finite number of such numbers for every base?
number-theory prime-numbers
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
$endgroup$
add a comment |
$begingroup$
For no particular reason I stumbled upon the sequence/set of all numbers where all substrings of the decimal representation is prime (A085823). It's quite easy to see that this must be a finite set. My approach was that the digits must alter between 3 and 7 eventually (since a prime can't end with 2 or 5 for larger than that and a prime can't consist of just two non-1 digits, those would be dividable by 2, 5 or 11) and we can see where it ends.
But if we change the base that approach fails. For example in base 8 a prime can end in 3, 5 or 7 AFAICS so such numbers can end in a altering sequence of 3, 5 and 7s but having tree digits to alter between there's always a way to vary the sequence.
I tried to write a python snippet to produce such numbers in different bases and for those I've tested the program seem to hang (ie don't seem to find any more numbers). Is it true that there are only a finite number of such numbers for every base?
number-theory prime-numbers
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
$endgroup$
For no particular reason I stumbled upon the sequence/set of all numbers where all substrings of the decimal representation is prime (A085823). It's quite easy to see that this must be a finite set. My approach was that the digits must alter between 3 and 7 eventually (since a prime can't end with 2 or 5 for larger than that and a prime can't consist of just two non-1 digits, those would be dividable by 2, 5 or 11) and we can see where it ends.
But if we change the base that approach fails. For example in base 8 a prime can end in 3, 5 or 7 AFAICS so such numbers can end in a altering sequence of 3, 5 and 7s but having tree digits to alter between there's always a way to vary the sequence.
I tried to write a python snippet to produce such numbers in different bases and for those I've tested the program seem to hang (ie don't seem to find any more numbers). Is it true that there are only a finite number of such numbers for every base?
number-theory prime-numbers
number-theory prime-numbers
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
asked Dec 21 '18 at 8:50
skykingskyking
14.3k1929
14.3k1929
add a comment |
add a comment |
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