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May 21

primeness of the digits in order for various bases

For what bases, is the list of digits in that base (not including the zero) as a number prime? For example, in base 10, the number 0123456789 is not prime (It is divisible by 9). In any even base, the equivalent of the base 10 divisiblity by 9 rule applies and additionally, in any odd base of the form 4n+1, there are an even number of odd numbers which are being multiplied by the power of the base (for example 1234 base 5 = 1*5^3+2*5^2+3*5+4) and is thus even. This leaves bases 4n+3. Base 3 (for which 12 base 3 = 5 base 10 prime). Base 7 (123456 base 7 =22875 base 10 divisible by 3 &5), Base 11 (123456789A base 11 = 2853116705 base 10 divisble by 5), Base 15 (123456789ABCDE base 15 = 2234152501943159 which is (according to Wolfram Alpha) composite with 7 and 113 as the smallest prime factors) and similarly on through at least base 35 which are all composite. (not sure how to express a 37 or a 38 in base 39 in wolfram alpha)

Is there any way to eliminate all of these showing that base 3 is the only one in which it is prime? Also, I believe the even base and 4n+1 base rules work for the numbers in descending order 987654321 base 10 and 4321 base 5 for example. For the descending order, 21 base 3 = 7 which is prime, and the rest seem to be composite.

It *appears* that for each of these bases, the 4n+3 numbers bases that the number both ascending and descending appear to be divisible by 2n+1, so for example, for base 15, both 123456789ABCDE and EDCBA987654321 are divisible by 7, but I'm not quite sure why and whether it is always true... Ideas? (note, this doesn't make base 3 really an exception since if 4n+3 = 3, 2n+1 = 1, which all primes are divisible by.)03:31, 21 May 2020 (UTC) — Preceding unsigned comment added by Naraht (talk • contribs)

Just to have a name for them, let's write L_n = 12...(n-1)_{base n}. A formula for L_n is (nⁿ-n²+n-1)/(n-1)². If n is of the form 4k+3 then n is divisible by both (n²+1)/2 and by (n-1)/2. To prove the first statement, note that nⁿ+n=n^4k+3-n=n((n²)^2k+1+1). For any x, x+1 divides x^p+1 when p is odd, so in this case n²+1 divides (n²)^2k+1+1 and therefore (n²+1)/2 divides the numerator (nⁿ-n²+n-1). It's not hard to show that (n²+1)/2 and n-1 are relatively prime, which implies that (n²+1)/2 divides L_n. To prove the second statement, let m=(n-1)/2, so n=2m+1 and L_n = ((2m+1)^2m+1-(2m+1)²+(2m+1)-1)/4m². To show m divides L_n is equivalent to showing 4m³ divides (2m+1)^2m+1-(2m+1)²+(2m+1)-1. Expand the first term by the binomial theorem to get 1+(2m+1)2m+(2m+1)m(2m)²+other terms which are all divisible by 4m³. When you add the remaining terms, namely -(2m+1)²+(2m+1)-1, everything cancels and you get L_n is congruent to 0 mod 4m³. Note that this work for any odd n, not just n=4k+3. I didn't look into the descending digits series too much; let's call them R_n, but you might notice that R_n = (n-2)L_n+n-1. Also note that the sequence L_n is OEIS: A023811. --RDBury (talk) 06:00, 21 May 2020 (UTC)[reply]

PS. R_n is OEIS: A051846. --RDBury (talk) 06:04, 21 May 2020 (UTC)[reply]

Can someone explain how this "integer repetition table" works?

Picture of table here. We're watching a Finnish TV show and this odd poster shows up in one scene and I've been trying to figure out just what it's for. I got one clue from reddit: "It seem to be a educational poster used in elementary school in early 20th century. There is a book published in 1904 explaining how to use the board called "Kokonaislukujen kertaustaulun selitys ja tulokset" by Hultin, Hj and Reinhard, Ph." Google translates the title to "Explanation and results of the integer repetition table". That's all I've been able to find.Cpergielx (talk) 23:08, 21 May 2020 (UTC)[reply]

This probably isn't be the best place to post this type of question; maybe somewhere specialized for Finnish language or culture would work better. But I'd say the elementary school idea seems plausible. According to Wiktionary, the words printed at the top of the posters mean subtraction table and addition table; wikt:yhteenlasku is addition, wikt:vähennyslasku is subtraction, and wikt:taulu in this context would mean table. The numbers seem too random to be memorized though, and the Finns do addition and subtraction in more or less the same way everyone else does, or at least the corresponding articles in the Finnish Wikipedia don't have any surprises. So I doubt it's some new method of arithmetic that only Finns know. I guess we could speculate all day on what it actually was, but without more information or the help of someone who grew up in Finland I don't see it going anywhere. --RDBury (talk) 00:13, 23 May 2020 (UTC)[reply]

Here are the tables.

Vähennyslasku-taulu.
-	O	P	R	S	T	U	V	Y	Ä	Ö	-
A	20	10	5	3	8	4	7	9	2	6	A
D	25	4	10	5	7	9	6	15	12	20	D
E	50	20	25	30	35	40	45	48	49	50	E
H	34	5	10	8	7	6	9	12	14	30	H
I	41	11	21	31	6	16	26	5	15	25	I
J	67	8	17	20	30	15	18	25	50	60	J
K	86	9	34	45	56	67	78	86	49	57	K
L	100	25	75	55	38	83	91	14	46	67	L
M	10	3.5	7.5	2.5	8.9	6.7	3.3	4.6	5.8	1.9	M
N	7.5	6.8	2.5	3.9	0.7	4.6	1.4	5.8	3.6	4.9	N
-	O	P	R	S	T	U	V	Y	Ä	Ö	-

Yhteenlasku-taulu.
+	O	P	R	S	T	U	V	Y	Ä	Ö	+
A	2	5	3	8	4	9	6	10	7	1	A
D	4	1	7	2	6	8	9	5	3	10	D
E	1	3	5	7	10	8	4	5	9	6	E
H	3	10	2	5	9	6	7	4	8	6	H
I	5	6	4	8	7	2	3	9	2	8	I
J	4	5	9	6	2	10	8	3	5	7	J
K	3	8	6	5	1	4	10	7	3	2	K
L	6	7	10	1	5	7	2	8	6	1	L
M	1	6	8	10	4	5	7	2	4	5	M
N	8	4	2	4	10	3	6	10	1	5	N
+	O	P	R	S	T	U	V	Y	Ä	Ö	+

Please feel free to prettify the formatting. catslash (talk) 02:12, 23 May 2020 (UTC)[reply]

Not anywhere near a solution, but a few remarks and observations. The image of the left table shows commas for the decimal separator – as one should expect for a tool used in Finnish education. This entry in an online Finnish catalogue describes the work Kokonaislukujen kertaustaulun selitys ja tulokset as an adaptation to Finland of a method due to Ph. Reinhard. WorldCat lists two works by Philipp Reinhard this could be based on: Neue Methode für den Rechnungsunterricht auf der Elementarstufe nebst einigen Tausend Übungsaufgaben (1874) and Methode für den Rechnungsunterricht (1890). In view of the 1904 date of the Finnish adaptation, the later work is the more likely source.

Both tables are 10×10 tables, so it is attractive to identify the row and column indices with the 10 decimal digits (0 to 9) or with the first 10 positive integers (1 to 10). However, if they have such a straightforward numerical significance, it is rather inexplicable that they are represented by letters (the "native" letters of the Finnish alphabet). I see no pattern, but in the left table there are many pairs of rows in which one row dominates another row. In particular, each of the rows E, J and K dominates each of the rows D, H, M and N, while E also dominates A and L dominates A, H, I, M and N. And the O column dominates all other columns. In a 10×10 table with random entries, the probability of even a single domination is less than 1 in 5. --Lambiam 21:59, 23 May 2020 (UTC)[reply]

Oooh, I found a review of the thing here. How's your Fraktur? --jpgordon^{𝄢𝄆 𝄐𝄇} 05:27, 24 May 2020 (UTC)[reply]

Maybe it's just a way to make exercises with somewhat random numbers without buying books. Making copies at the school probably wasn't an option at the time. Put up the board and ask the class to compute row E minus row H, and so on. PrimeHunter (talk) 22:54, 23 May 2020 (UTC)[reply]

This would explain the large number of dominations in the subtraction board: it gives the teacher 18 subtraction exercises that do not involve negative numbers – a concept that may not yet have been introduced to the students at that stage. For addition this is not an issue, so it is not surprising the addition board is domination-free. There may have been several such boards; the entries in the addition board in the photograph are simple and may have served for the first addition exercises. Presumably there were then also similar multiplication boards. --Lambiam 02:00, 24 May 2020 (UTC)[reply]

The review in Fraktur of Reinhard's 1874 book found by jpgordon describes an (apparently) earlier form, but indeed a way to make exercises as surmised by PrimeHunter. According to the description, the boards, sized one square metre, display a 9×9 matrix whose entries are the numbers 1 through 9, presented in white on a dark background. Note that Finnish taulu can also mean "picture", "print" etc., especially a framed one, and "board", as for example a billboard, or blackboard. --Lambiam 10:51, 24 May 2020 (UTC)[reply]

From the review: "Mr. Reinhard now tells us a very simple procedure to demonstrate as many and different exercises in arithmetic with pure numbers to all the students through all basic operations up to the unlimited number range. His method is based on a scheme consisting of 81 squares, in which the basic numbers 1 to 9 are set in nine vertical and just as many horizontal rows so that the first two are never repeated in two successive numbers. Various combinations and <verwechslungen> result in many thousands of exercise examples..." Mostly a Google translate of my transcription of the blackletter text. Not sure what "verwechslungen" should translate to idiomatically. --jpgordon^{𝄢𝄆 𝄐𝄇} 16:16, 24 May 2020 (UTC)[reply]

Verwechslung literally means "change", "substitution" or "replacement" (of one thing by another); idiomatically it usually refers to a mistaken identification, but here I think it means "a swap", and since any permutation is possible through a sequence of swaps, the translation "permutations" for the plural seems reasonable. --Lambiam 20:36, 24 May 2020 (UTC)[reply]

If I was given that description as a rough translation without the context of Cpergielx's images, then I'd figure Mr. Reinhard was suggesting worksheets (worktablets?) such as:

+ 3 6 4 7 2 8 1 5 9 +

5 5

9 9

7 7

1 1

6 6

3 3

8 8

4 4

2 2

+ 3 6 4 7 2 8 1 5 9 +

giving the student 81 sums to complete, with the "thousands of exercise examples" coming from the "various combinations and permutations" of row and column headers. Here I interpret "so that the first two are never repeated in two successive numbers" as meaning not permitting adjacent consecutive integers in the row and column headers, since having such makes it easier for the student to compute a sum as an increment or a product as a sum. This loose interpretation seems within the realm of possibility given a rough translation of a colloquial description of a mathematical construct, but it would then appear to have no bearing on the mystery tables.

Chuck, the Redditor who mentioned Mr. Reinhard's work didn't have a definitive link between it and your tables, did they? And are those photos actual sreenshots from Hooked? Their quality seems too high. -- ToE 15:07, 28 May 2020 (UTC)[reply]