Prime Decomposition of Repunits

Theorem

The prime decomposition of the repunits is as follows:

$\ds R_2$

$\ds 11$

$\ds R_3$

$=$

$\ds 3 \times 37$

$\ds R_4$

$=$

$\ds 11 \times 101$

$\ds R_5$

$=$

$\ds 41 \times 271$

$\ds R_6$

$=$

$\ds 11 \times 111 \times 91$

$\ds $

$=$

$\ds 3 \times 7 \times 11 \times 13 \times 37$

$\ds R_7$

$=$

$\ds 239 \times 4649$

$\ds R_8$

$=$

$\ds 1111 \times 10 \, 001$

$\ds $

$=$

$\ds 11 \times 73 \times 101 \times 137$

$\ds R_9$

$=$

$\ds 111 \times 1 \, 001 \, 001$

$\ds $

$=$

$\ds 3^2 \times 37 \times 333 \, 667$

$\ds R_{10}$

$=$

$\ds 11 \times 11 \, 111 \times 9091$

$\ds $

$=$

$\ds 11 \times 41 \times 271 \times 9091$

$\ds R_{11}$

$=$

$\ds 21 \, 649 \times 513 \, 239$

$\ds R_{12}$

$=$

$\ds 111 \times 1111 \times 900 \, 991$

$\ds $

$=$

$\ds 3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$

$\ds R_{13}$

$=$

$\ds 53 \times 79 \times 265 \, 371 \, 653$

$\ds R_{14}$

$=$

$\ds 11 \times 1 \, 111 \, 111 \times 909 \, 091$

$\ds $

$=$

$\ds 11 \times 239 \times 4649 \times 909 \, 091$

$\ds R_{15}$

$=$

$\ds 111 \times 11 \, 111 \times 90 \, 090 \, 991$

$\ds $

$=$

$\ds 3 \times 31 \times 37 \times 41 \times 271 \times 2 \, 906 \, 161$

$\ds R_{16}$

$=$

$\ds 11 \, 111 \, 111 \times 100 \, 000 \, 001$

$\ds $

$=$

$\ds 11 \times 17 \times 73 \times 101 \times 137 \times 5 \, 882 \, 353$

$\ds R_{17}$

$=$

$\ds 2 \, 071 \, 723 \times 5 \, 363 \, 222 \, 357$

$\ds R_{18}$

$=$

$\ds 111 \, 111 \, 111 \times 1 \, 000 \, 000 \, 001$

$\ds $

$=$

$\ds 3^2 \times 7 \times 11 \times 13 \times 19 \times 37 \times 52 \, 579 \times 333 \, 667$

$\ds R_{19}$

$=$

$\ds 1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$

$\ds R_{20}$

$=$

$\ds 1111 \times 11111 \times 900 \, 099 \, 009 \, 991$

$\ds $

$=$

$\ds 11 \times 41 \times 101 \times 271 \times 3541 \times 9091 \times 27 \, 961$

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Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,111,111,111,111,111,111$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,111,111,111,111,111,111$