for which values of p this expression has the form s*2^k?

[enzocreti]

let be p an arbitrary prime...
can you find values of p such that


(10^p-1)/9+1 has the form s*2^k?


where s is a prime and k an integer >0


example p=3 does the job
infact (10^3-1)/9+1 has the form 7*2^4


p=5 does not the job because


(10^5-1)/9+1 has the form 2^3*3*463...

[enzocreti]

I found that the primes that do the job up to p=103 are: 3, 7, 43.


Is there some relation with the fact that


10^r+333667 is prime for r=3,7 and 43 and no other prime r up to r=26.000?

[wblipp]

Originally Posted by enzocreti

let be p an arbitrary prime...
can you find values of p such that


(10^p-1)/9+1 has the form s*2^k?

2, 3, 7, 229, 1579

[Dr Sardonicus]

I note that

if n > 3 then N = ((10^n - 1)/9 + 1)/8 is an odd integer.

Also, n need not be prime for N to be prime.

? for(i=4,1000,n=1+(10^i-1)/9;n=n/8;if(ispseudoprime(n),print(i)))
4
7
16
43
58
106
160
229
628