A Poulet-number is a composite positive integer $\ N>1\ $ satisfying the condition $$2^{N-1}\equiv 1\mod N$$ The numbers $$10^{16}+8\ 663\ 854\ 653$$ and $$10^{17}+209\ 045\ 665\ 633$$ are Poulet-numbers.
Are the numbers the smallest Poulet-numbers above $10^{16}$ and $10^{17}$ respectively ?
How can I search the smallest Poulet number above a given number efficiently ?
A tabulation of all the Poulet numbers less than $2^{64}(\approx1.8447\times10^{19})$ can be found here. There we find that the answer to your first question is no:
Your number $10^{16}+\hbox{8 663 854 653}$ is the third Poulet number beyond $10^{16}$; the first two are $10^{16}$ plus $\hbox{6 286 491 369}$ and $\hbox{8 250 001 701}$, respectively.
And your Poulet number $10^{17}+\hbox{209 045 665 633}$ is in fact the eighth Poulet number beyond $10^{17}$. The first seven are $$10^{17}+(10\ 102\ 756\ 401, 56\ 076\ 920\ 001,116\ 526\ 957\ 249,139\ 941\ 938\ 721,150\ 553\ 089\ 531, 157\ 387\ 600\ 201, \rm{and}\ 174\ 032\ 555\ 661),$$ respectively.