User:FUNG Cheok Yin/proof(i) A04290

This item has been flagged for DELETION.

wrong title

Discarded items are items flagged for deletion to facilitate the maintenance of the wiki. If an item (a category page, an article page, an extras page, a help page, a template page, a file page, ...) is created erroneously (wrong namespace, wrong naming convention, wrong spelling, wrong capitalization, wrong grammatical number (singular/plural), and so on...) only a sysop or an administrator has the proper rights to delete it, but we may facilitate their task by flagging the item as discarded.

To categorize an item as Category:Discarded items you just use the template tag {{discarded|Reason for discarding}} in the item's (a category page, an article page, an extras page, a help page, a template page, a file page, ...) page, where "Reason for discarding" is a short explanation of the request for deletion.

Proof of A04290(10^m - 1) = repunit(9 * m)

link to seq A04290

Let g be the smallest required multiple of 10^m - 1 , i.e. g = A04290(10^m - 1).

   10^m - 1 = repunit(9 * m)

Let c be the odd number such that g = c * repunit(m) .

Concerning the last digit of g, c must end with 1. For m >= 2, concerning c ≡ 1 (mod 10), and g is only allowed to be consisted of digit 1 or 0, the second last digit of c must be 0, i.e. c ≡ 1 (mod 100). For m >= 3, concerning c ≡ 1 (mod 100) and the restriction above, we would have the third last digit of c must be 0, i.e. c ≡ 1 (mod 1000) . Similar arguments proceed.

Then we have:

   g = (1 + 10^s_1 + 10^s_2 + ... + 10^s_r) * repunit(m)

Let s_0 = 1.

Without loss of generality, we require s_{i+1} - s_{i} >= m for 0 <= i < r, otherwise there will be place(s) overlapped during addition and then digits besides 0 and 1 will present on the sum.

   g = c / 9 * 9 * repunit(m) = (c / 9) * (9 * repunit(m))

c must be divisible by 9.

By the divisibility rule of 9 (digitsum rule), r+1 must be divisible by 9.

The smallest combination of s_i, for which r+1 is divisible by 9, is s_i = i*m for 0 >= i >= r = 8.

Hence

   g = repunit(9 * m)

Check again against the requirement, It appears that such g is consisting of 1's only, and a multiple of 10^m - 1.

Conclusion:

   a(10^m - 1) = repunit(9 * m) . □

---

Finish on Dec 24 2020