A064539 All...are divisible by 3 but why?

[David Harden]

>From: zak seidov <zakseidov at yahoo.com> 
>To: "Pfoertner, Hugo" <Hugo.Pfoertner at muc.mtu.de>, >seqfan at ext.jussieu.fr 
>Subject: RE: A064539 All...are divisible by 3 but why? Date: Fri, 24 Jun >2005 02:21:18 -0700 (PDT) 
>
>Dear Hugo, 
>     seqfans, 
>
>the (one) point is that 
>other day Neil explained me 
>that "base" means that 
>SEQ is not so much interesting for OEIS. 

The converse is not true, though, as a sequence can be uninteresting without have a base-dependent definition. 
 	 	
>i'm afraid that i don't yet understand 
>the "real" meaning of this base keyword "base"... 
>silly(?) Q: 
>is any number that is prime in base 10 
>is prime in other bases as well? 

The definition of prime is independent of the base, since it is given purely in terms of the definition of multiplication of positive integers. That is defined without respect to a base: if a is the cardinality of the set A and b is the cardinality of the set B, then a*b is the cardinality of the Cartesian product set A x B, which is the set of ordered pairs where the first entry in the ordered pair is in A and the second entry is in B. The string of digits "19" represents a prime in base 10 but not in base 16. They are not both the number nineteen; "19" represents the number nineteen when read as a base-ten string. When read as a hexadecimal string, it represents the number twenty-five. Please learn that numbers are more than the strings of symbols used to represent them. 
 	 	
>the more(?) interesting Q is: 
>are really cases n=1 and 28 
>the only two(?) ones(??) 
>making 3^(2n)+(2n)^3 prime? 
>see A109215 (pending, and, sorry, again "base"), 
>thanks, zak

This question is definitely more interesting, and my personal guess is that there are infinitely many n such that 3^(2n) + (2n)^3 is prime. How far has this sequence been computed? It is interesting that the residue of this quantity modulo the prime p will require the knowledge of p modulo the binomial coefficient (p  2) whenever 3 is a primitive root modulo p or when p == 11 (modulo 12) and 3 has order (p-1)/2 modulo p.

>--- "Pfoertner, Hugo" wrote: 

>> 
>> Recently we had a discussion on the use of the 
>> "base" keyword. In my opinion 

and mine too, for the same reason 
>> the sequence above is an example where "base" is 
>> totally inappropriate, 
>> because the result has nothing to do with the 
>> respresentation of the numbers 
>> in a certain base-x system. 
>>
>> Hugo 
>>