A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$. ec numbers are introduced obtained by the concatenation of two consecutive Mersenne numbers (40952047 for example). Ec(7)=12763 and ec(8)=255127 are both congruent to 7 mod 1063. I did not find yet another example of ec(k) and ec(k+1) both congruent to 7 mod 1063. Is there any particolar mathematical reason, can be that ruled out or is it just coincidence?
Asked
Active
Viewed 53 times
1 Answers
1
The numbers
$ec(289922)$ and $ec(289923)$ are both congruent to $7$ modulo $1063$
Peter
- 86,576
-
$k=2268439$ is the third solution – Peter Nov 18 '18 at 19:24
-
$289922=2*144961$. $144961$ is the concatenation of two squares 12^2 and 31^2. – Nov 19 '18 at 14:20