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For any base, its repunits of length 0 and 1 are squares (02 and 12). And if b = a2 − 1, the base-b repunit of length 2 equals a2 and so is a square. Other than that, square repunits appear to be rare. The base-2 repunits are the Mersenne numbers. Since Mn ≡ 3 (mod 4) for n > 1 while a2 ≡ 0 or 1 (mod 4), we see that no further squares can be found there. But we have 111113 = 121 = 112 and 11117 = 400 = 202, so square repunits of length greater than one are not necessarily impossible. Are there other non-trivial square repunits? -- Lambiam 08:17, 10 August 2021 (UTC)
Mathematics desk | ||
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< August 9 | << Jul | August | Sep >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
For any base, its repunits of length 0 and 1 are squares (02 and 12). And if b = a2 − 1, the base-b repunit of length 2 equals a2 and so is a square. Other than that, square repunits appear to be rare. The base-2 repunits are the Mersenne numbers. Since Mn ≡ 3 (mod 4) for n > 1 while a2 ≡ 0 or 1 (mod 4), we see that no further squares can be found there. But we have 111113 = 121 = 112 and 11117 = 400 = 202, so square repunits of length greater than one are not necessarily impossible. Are there other non-trivial square repunits? -- Lambiam 08:17, 10 August 2021 (UTC)