Prime Decomposition of Repunits

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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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