Corrected On sequence nXn where x equal 09,18,27,36,45,54,63,72,81,90

[f.firoozbakht at sci.ui.ac.ir]

Dear Karima,

> For example there are no prime on the form p07p 
> (p between 999973 and 10000017 ).
> 99997307999953
> 100000170710000017
> there are similar for 14,21, ...
> 

That is because If  10^6-1 < n < 10^7 then  
n.07.n = n*10^(7+2)+7*10^7+n=7*10^7+n*(10^9+1)  and 
since 7 divides 10^9+1 ,so 7 divides n.07.n .
Hence for n between 10^6 and 10^7  n.07.n is composite.
This is true also for n.14.n, n.21.n, ..., n.98.n

By Fermat theorem we have 10^(6k+3)+1 == 0 ( mod7 ) 

so we can conclude:

If  10^(6k)-1 < n < 10^(6k+1) and m is a multiple of 7 with 
two digits then 7 divides  n.m.n   so in this case n can't 
be prime.
Hence if n be in the following intervals ;
 
[10^6-1,10^7], [10^12-1,10^13], [10^18-1,10^19],... 

and m= 07,14,21,...,98   then  n.m.n is composite.

we have similar statement for other cases (m has more digits or...).

Regards

Farideh




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