It is a question I always ask when I was a primary student:

Could a number only contains digit’1′ with $ p$ digits writes $ $ A=\frac{10^{p}-1}{9} = \underbrace{11\cdots 1}_{p\text{ digits}}$ $ be a prime?

When $ p$ is not a prime, the factorization of $ A$ is obvious, but if $ p$ is a prime, the factorization of $ A$ may be subtle.

Here I made a list of some result for prime $ p$

$ $ \begin{array}{c|lcr} p & \ \text{factors} \ \hline 3 & 3 \times 37\ 5 & 41 \times 271 \ 7 & 239 \times 4649 \ 11 & 21649 \times 513239 \ 13 & 53 \times 79 \times 265371653 \ 17 & 2071723 \times 5363222357 \ 19 & \text{prime} \ 23 & \text{prime} \ 29 & 3191 \times 16763 \times 43037 \times 62003 \times 77843839397 \ 31 & 2791 \times 6943319 \times 57336415063790604359 \ 37 & 2028119 \times 247629013 \times 2212394296770203368013 \ 41 & 83 \times 1231 \times 538987 \times 201763709900322803748657942361 \ 43 & 173 \times 1527791 \times 1963506722254397 \times 2140992015395526641 \ 47 & 35121409 \times 316362908763458525001406154038726382279 \ 53 & 107 \times 1659431 \times 1325815267337711173 \times 47198858799491425660200071 \ 59 & 2559647034361 \times 4340876285657460212144534289928559826755746751 \ 61 & 733 \times 4637 \times 329401 \times 974293 \times 1360682471 \times 106007173861643 \times 7061709990156159479 \ 67 & 493121 \times 79863595778924342083 \times 28213380943176667001263153660999177245677 \ \end{array} $ $

I am not major in number theory, just curious again if there is any theorem or former work on this kind of question.