Is $$1\underbrace{0101\cdots 01}_{k\text{ 01}}$$ composite for all $k \geq 2$?

when $k=2,3$ . they are composite numbers.

$$10101 = 3\times 7\times 13\times 37$$

$$1010101 = 73\times 101\times 137$$

  • 2
    $\begingroup$ Are you asking whether or not there is a prime number of this form? Pretty easy to check that they aren't all prime. $\endgroup$
    – lulu
    Sep 2, 2016 at 12:40
  • $\begingroup$ @DanielR I guess number in this pattern is not a prime. : ) $\endgroup$
    – Laura
    Sep 2, 2016 at 12:43
  • $\begingroup$ @lulu yes. I think they are all composite numbers. $\endgroup$
    – Laura
    Sep 2, 2016 at 12:44

1 Answer 1


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 $99$.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .