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-   -   for which values of p this expression has the form s*2^k? (https://www.mersenneforum.org/showthread.php?t=24279)

enzocreti 2019-04-10 14:17

for which values of p this expression has the form s*2^k?
 
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 2019-04-10 14:57

Primes that do the job up to 103
 
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 2019-04-11 01:14

[QUOTE=enzocreti;513344]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?[/QUOTE]

2, 3, 7, 229, 1579

Dr Sardonicus 2019-04-11 14:01

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.
[code]? 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[/code]


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