Repunit

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The term repunit comes from the words 'repeated' and 'unit;' so repunits are positive integers in which every digit is one. (This term was coined by A. H. Beiler in [Beiler1964].) For example, R₁=1, R₂=11, R₃=111, and R_n=(10ⁿ-1)/9. Notice R_n divides R_m whenever n divides m.

Repunit primes are repunits that are prime. For example, 11, 1111111111111111111, and 11111111111111111111111 (2, 19, and 23 digits). The only other known repunit primes are the ones with 317 digits: (10³¹⁷-1)/9, 1,031 digits and (10¹⁰³¹-1)/9.

During 1999 Dubner discovered R₄₉₀₈₁ = (10⁴⁹⁰⁸¹-1)/9 was a probable prime. In October 2000, Lew Baxter discovered the next repunit probable prime is R₈₆₄₅₃. In 2007 the probable primes R₁₀₉₂₉₇ (Bourdelais and Dubner) and R₂₇₀₃₄₃ (Voznyy and Budnyy) were found. In 2021 the probable primes R_5794777 and R_8177207 were found (Batalov and Propper). It will be some time before these giant primes are proven! As the poet wrote:

Ah, but a man's reach should exceed his grasp, or what's a heaven for? (Robert Browning)

What makes these repunits R so difficult to prove prime is that we do not have an easy way to factor R-1 or R+1. The (practically) quick methods of primality proving all require factoring.

Even though only a few are known, it has been conjectured that there are infinitely many repunit primes. To see why just look at the graph of the known repunit primes and probable primes (here we graph log(log(R_n)) verses n.