Is $1\underbrace{0101\cdots 01}_{k\text{ 01}}$ composite for all $k \geq 2$? Is $$1\underbrace{0101\cdots 01}_{k\text{ 01}}$$ composite for all $k \geq 2$?
when $k=2,3$ . they are composite numbers.
$$10101 = 3\times 7\times 13\times 37$$
$$1010101 = 73\times 101\times 137$$

None of these are prime.  To see that we write your numbers as $$S_k=1+10^2+10^4+\cdots+10^{2k}=\frac {10^{2k+2}-1}{10^2-1}=\frac {(10^{k+1}+1)(10^{k+1}-1)}{99}$$
To conclude we remark that for large $k$ it is clear that both factors in the numerator are larger than $99$.
