Consider summing the digits of the number 141. We have 1 + 4 + 1 = 6. You might remember the rule that if the "sum of the digits" of some integer N is divisible by 3, then the original integer is also divisible by 3 (in the case 141 = 47 x 3).......
Now instead of determining divisibility by 3, consider the problem of determining the primality of the "sum of the digits" of some integer N that is written in base B.
In particular, consider base B = 7. Under this base, the integer 7 will be represented as 10. The sum of its digits will be 1 + 0 = 1.
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Prove that if an integer N is prime and is written in base 7, then the sum of its digits is always prime (excluding the example 7 above). If it is not true, find a counter-example......
The primes up to 101 written in base 7 are as follows;
2
3
5
10
14
16
23
25
32
41
43
52
56
61
65
104
113
115
124
131
133
142
146
155
166
203
If you add the digits in each of these integers, you get a prime number!
I suspect there is a counter example higher up, but I haven't been able to download a program like PARI or something else that will let me test millions of numbers very quickly. Keep in mind that this algorithm can only construct smaller prime integers from bigger ones.
It is true even for some composite numbers
2875 (base 10)=11245(base 7)
The sum of digits is 13 which is prime but 2875 is NOT prime.
So it is not helpful.
It is like saying that if n>2 is prime then n is ALWAYS odd.
Nothing new HERE. The prime number need some very deep tool to reveal its secrets.