Corrected On sequence nXn where x equal 09,18,27,36,45,54,63,72,81,90
f.firoozbakht at sci.ui.ac.ir
f.firoozbakht at sci.ui.ac.ir
Mon Mar 1 03:31:42 CET 2004
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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