Wednesday 26 October 2022

#26 Repunit and Residual... from old Math Terms notes

Repunit The term repunit, for a number made up of all unit digits, was created by Albert Beiler in the 1960's as a contraction of repeated unit. It still does not appear in most dictionaries, and is therefore difficult to trace... more to come, I hope.

One of the most famous puzzle masters of the early 1900's was Henry Ernest Dudeney. In his puzzle book,The Canterbury Puzzles, of 1907 he poses a problem about Repunit factoring, without using the term repunit. The problem is posed thus:

"It used to be told at St Edmondsbury," said Father Peter on one occasion, "that many years ago they were so overrun with mice that the good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, and the total was 1,111,111 mice. How many cats do you suppose there were?"
Later the problem provides that there is more than one cat, and each cat kills more than one mouse, in fact each cat killed more mice than there were cats.

I will begin to discuss the answer shortly, so those who wish to solve the problem first might stop reading and take a few moments to work on a solution.
Dudeney points out that for the solution to be unique, there must be only two factors of 1,111,111. In truth, the factors of 1,111,111 are 239 and 4649, so there must have been 239 cats who each caught 4649 mice.

In the solutions in the second edition in 1919, Dudeney discusses the general problem of finding solutions of repunits (again without using the term) and gives some interesting tables and remarks which I have tried to recopy accurately

Lucas, in his L'Arithmetique Amusante, gives a number of curious tables which he obtained from an arithmetical treatise, called the Talkhys, by Ibn Albanna, an arabian mathematician and astronomer of the first half of the thirteenth century. In the Paris National Library are several manuscriptes dealing with the Talkhys, and a commentary by Alkalacadi, who died in 1486. Among the tables given by Lucas is one giving all the factors of numbers of the above form (repunits) up to n=18 (eighteen ones in a row). It seems almost inconceivable that Arabians of that date could find the factors where n=17 as given in my introduction [On page 18 of the introduction he gives the factors of 11,111,111,111,111,111 as 2,071,723 and 5,363,222,357.]. But I read Lucas as stating that they are given in Talkhys, though an eminent mathematician reads him differently, and suggest to me that they were discovered by Lucas himself. This can, of course, be settled by an examination of the Talkhys, but this has not been possible during the war.(WWI)

The difficulty lies wholly with those cases where n is a prime number. If n=2, we get the prime 11. The factors when n=3, 5, 11, and 13 are respectively (3x37), (41 x 271), (21,649x513,239), and (53 x 79 x 265,371,653)[Dudeney used a raised dot for multiplication which I have replaced with an x for my convenience]. I have given in these pages the factors where n=7 and n=17. The factors where n=19, 23, and 37 are unknown, if there are any.* [emphasis added]

In a footnote He points out that "Mr. Oscar Hoppe, of New York, informs me that, after reading my statement in the introduction [where he says n=19 is prime], he was led to investigate the case of n=19, and after long and tedious work he succeeded in probing the number to be a prime. He submitted his proof to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. He refers me to the Proceedings of the Society for 14th February, 1918.

Dudeney also points out three "curious series of factors" that he thought would "doubtless interest the reader." They are shown here: n=2 --> 11
n=6 --> 11 x 111 x 91
n=10 --> 11 x 11,111 x 9091
n=14 --> 11 x 1,111,111 x 909,091
Can you guess n=18?

Or the same numbers can be written as
n=6 --> 111 x 1,001
n=10 --> 11,111 x 100,001
n=14--> 1,111,111 x 1,000,000,001

For the cases where n is a multiple of 4, we get
n=4 --> 11 x 101
n=8 --> 11 x 101 x 10,001
n=12--> 11 x 101 x 100,010,001
n=16--> 11 x 101 x 1,000,100,010,001

Residual Sit back, stay right there, and I will tell you the origin of residual. Wait! I just did. The common re prefix means back and sid is from the Latin sedere which means to sit, so the literal meaning of residual is one who sits back or, more appropriately, stays seated. In statistics we use it in the same sense as residue, that which remains (stays seated) when something else is taken away; what remains from the observed amount when the predicted amount is removed. Another closely related word is residence. Other words drawn from the sedere root include sedentary [one who sits around a lot], sediment [stuff that settles], and sedative [something that keeps you from moving around].

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