CC BY
32
Труды БГТУ, 2022, серия 3, № 1, с. 15-18
15
УДК 511.172
Thiha Bo
Mandalar University (Myanmar)
THE VISUALIZE FORMULATION OF DIVISIBILITY
For the purposes of cryptography it is necessary to develop effective methods and algorithms: to check the simplicity of integers; to find large prime numbers; to factorize integers.
This paper studies a generalized method for constructing algorithms to check the divisibility of integers by a given number b in various number systems by analyzing the sets of divisors of base (a given number b), prebase (number b - 1), and postbase (number b + 1). It is indicated that the rules for testing divisibility by a given number may have different complexity depending on the number system used.
The paper introduces the formulations of some theorems with proofs. The theorems are supported by concrete examples.
These theorems can formulate for many divisibility rules for any number over the any base. Some numbers are although difficult over base 10, they are easy over another base. Some numbers, such as primes, have direct rules, but some composites have combined rules.
Key words: base factors, prebase factors, postbase factors, rise, visualize array.
For citation: Thina Bo. The visualize formulation of divisibility. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2022, no. 1 (254), pp. 15-18.
Тиха Бо
Университет Мандалая (Мьянма)
ВИЗУАЛИЗИРУЕМАЯ ФОРМУЛИРОВКА ДЕЛИМОСТИ
Для целей криптографии необходимо разрабатывать эффективные методы и алгоритмы: проверки простоты целых чисел; поиска больших простых чисел; факторизации целых чисел.
В данной статье изучается обобщенный способ построения алгоритмов проверки делимости целых чисел на заданное число b в различных системах счисления путем анализа множеств делителей базы (заданного числа b), предбазы (числа b - 1) и постбазы (числа b + 1). Указано, что правила проверки делимости на данное число могут иметь разную сложность в зависимости от используемой системы счисления.
В статье приводятся формулировки некоторых теорем с доказательствами. Теоремы подкреплены конкретными примерами. Эти теоремы позволяют сформулировать множество правил делимости для любого числа в любой системе счисления. Некоторые числа, хотя и являются сложными при делении в системе счисления по основанию 10, легко делятся в системе счисления по другому основанию.
Ключевые слова: делители базы, делители предбазы, делители постбазы, рост, визуализация массива.
Для цитирования: Тиха Бо. Визуализируемая формулировка делимости // Труды БГТУ. Сер. 3, Физико-математические науки и информатика. 2022. № 1 (254). С. 15-18.
Introduction. In number theory, the properties of integers are studied. In this paper, a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits, were developed generally. For base 10 (decimal system), Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American. In this paper, the divisibility rules for any number, any base and the best formulae are derived by visualization. Today, all of the calculations are calculated by electronic devices. But these devices are made by human. If the algorithms of the calculation of the devices are reduced to simplest way by using the theorems in this paper, it will be great benefit for us [1-3].
Main part. To get the sense of divisibility, some definitions and notations are defined and introduced.
The properties of numbers over any base are characterized.
The prime divisors or the power of prime divisors of a base b is called the base factors of b.
Bb is the notation of the set of all base factors of b. For examples: B10 = {2, 5}.
Suppose the digit of base 8 are order triple pair of zero and one, i. e.
0 = 000, 1 = 001, 2 = 010, 3 = 011, 4 = 100, 5 = 101, 6 = 110, 7 = 111, 8 = 1000, B8 = {010, 100, 1000} = {2, 4,8}.
The divisors of b - 1 is called the prebase factors of b. Pb is the notation of the set of all prebase factors of b. For examples:
P10 = {3, 9}, P8 = {111} = {7}.
Трулы БГТУ Серия 3 № 1 2022
The divisors of b + 1 is called the postbase factors of b. Qb is the notation of the set of all postbase factors of b. For examples:
Q10 = {11}, QS = {011, 1001} = {3, 9}.
The different form one's digit of a multiply of number n to one ' s digit of another multiply n is called run of n over base b. The different between ten's digit of a multiply of number n and ten's digit of another multiply n is called rise of n over base b. rb(p, q) n y : X is the notation of the simplest integral ratio of rise y and run x ofpn and qn over base b, where x ^ 0 and y 0. Rb(n) is the notation of the set of all ratios rb(p, q)n for n and base b. For examples:
For 7 and 14 over base 10, run is -3 and rise is 1, i. e., r10(1, 2)7 = 1 : -3.
For 7 and 21 over base 10, run is -6 and rise is 2, i. e., r10(1, 3)7 = 1 : -3.
For 7 and 28 over base 10, run is 1 and rise is 2, i. e., r10(1, 4)7 = 2 : 1.
For 7 and 35 over base 10, run is -2 and rise is 3, i. e., r10(1, 5)7 = 3 : -2. Then
R10(7) = {-1 : 3, 2 : 1, -3 : 2, -4 : 5, 1 : 4, -6 : 1, ...}.
Similarity
R10(9) = {-1 : 1, -10 : 1, 8 : 1, 7 : 2, ...}, R10(11) = {1 : 1, 12 : 1, -10 : 1, -9 : 2, ...},
R10(3) = {-1 : 1, -10 : 1, 8 : 1, 7 : 2, ...}.
The element of Rb(n) is defined as the best coefficient or best ratio of n over base b, if its denominator 1 and the modulus of numerator is minimum. For examples: 2 : 1 is the best coefficient of 7 over base 10. 1 : 1 is the best coefficient of 11 over base 10. -1 : 1 is the best coefficient of 3 and 9 over base 10.
The best coefficient of a prebase factor of a base b is always -1 : 1 and the best coefficient of a postbase factor of a base b is always 1 : 1.
Lemma. If y : x e Rb (n), then n | x + by.
Proof. By definition, y : x = rb(p, q)n.
Let pn = ab + P and qn = yb + S.
So, x = S - P and y = y - a.
x + by = S - P + b(y - a) =
= yb + S - (ab + P) = = qn — pn = (q - p)n.
Depend on the characteristics of a number over a given base, the divisibility rules were different each other. But we need only four theorems which are developed generally as follow.
Base Factor Theorem. If n is a base factor of b and
p
m = Z aibi = a0b0 + axbx +... + apbp, then n | m if and
i =0
only if n | a0.
Proof. Since n is a base factor of b, n | b. The result is obviously.
Prebase Factor Theorem. If n is a prebase factor
p
of b and m = Zaibi = a0 + axb +... + apbp, then n | m
i=0
p
if and only if n | Z ai.
i=0
Proof. Since n is a prebase factor, n | b - 1. Since b - 1 | 1 - b, for i = 1, 2, 3, .... So, n | (1 - b)ai + ... + (1 - bp)ap and let (1 - b)ai + + ... + (1 - bp)ap = sn.
Suppose n | m. Let m = kn.
p
Z a = ao + a1 +...+ap =
i=0
= a0 + aj +... + ap + a1b +... + apbp - a1b -... -
-apbp = a0 + a1b +... + apbp + (1 - b)aj +... +
+(1 - bp) ap = kn + sn = (k + s)n.
p
m = Z aibi = a0 + a1b +... + apbp = i=0
= a0 + axb +... + apbp - aj -... - ap + aj +... + ap =
= a1(b -1) +... + ap(bp -1) + a0 + a1 +... + ap = = -sn + tn = (t - s)n.
Postbase Factor Theorem. If n is a postbase fac-
p
tor of b and m = Zaibi = a0 + a1b +... + apbp, then
i=0
p
n | m if and only if n | Z (-1); ai.
i=0
Proof. Since n is a postbase factor, n | b + 1. Since b + 1 | 1 + b', for i = 1, 3, 5, ... and b + 1 | 1 - bi, for i = 2, 4, 6, ....
So, n | -a:(1 + b) + a2(1 - b2) - a3(1 + b3) +.+ + ap((-1)p - bp) = sn.
Suppose n | m. Let m = kn.
Z (-1)^. = a0 - a, +... + (-1)pap =
i=0
= a0 -a1 + a2 -a3 +... + (-1)pap + a1b + a2b2... +
+ apbp - ab -... - apbp =-ax(1 + b) + + a2(1 -b2) -a3(1 + b3) +... + ap((-1)p -bp) = = kn + sn = (k + s)n.
p
Conversely, suppose n | Z (-1) iai. Let
i =0
Z (-1) ial = tn.
i=0
TpyAbi BrTy Cepnn 3 № 1 2022
Thiha Bo
17
m = ^ aibi = a0 + aAb +... + apbp = a0 + «jb +
i=0
+... + apbp -al + a2 -a3 +... + (-1)p ap + ax -a2 +
+ a3 -... - (-1)p ap = a0 - aj + a2 - a3 +... +
+ (-1) pap + aj(b +1) - a2(1 - b2) + ^(1 + b3) +
+... - ap ((-1)p - bp) = tn - sn = (t - s)n.
Visualize Divisibility Theorem. If y: x e Rb (n), (n, y) = 1, (n, b) = 1, m = bp + q, then n | m if and only if n | qy - xp.
Proof Let, by Lemma, x+by=for some integer Suppose n | m, i. e., bp + q = An, for some integer A.
bpy + qy = Any;
bpy + xp - xp + qy = Any;
(x + by)p + (qy - xp) = Any;
|np + (qy - xp) = Any;
qy - xp = (Ay - |p)n.
Conversely, suppose n | qy - xp, i. e., qy - xp = = kn, for some integer k.
|np + qy - xp = |np + kn; (x + by)p + qy - xp = (|p + k)n; xp + byp + qy - xp = (|p + k)n; (bp + q)y = (|p + k)n.
Since (n,y) = 1, n | bp + q, i. e. n | m. Convergence of Formulae. Using the best coefficient of a number n of a base b, and using the Visualize Divisibility Theorem, we will have a convergence formula for divisibility of the number n. For a number n belong to none of the set Bb, Pb or Qb and (n, b) ^ 1, using combination formula of the formulae of its prime and power of prime factors. Some Visualize Array and Some Examples. 1) Finding ratio of 7 over base 10 on visualize array of base 10.
Table 1
The visualize array of №o(7) = {2 : 1, ...}
Note. Negative (gray) and positive (boundary).
2) Some formulae of 7 over base 10.
There were many convergence formulae of 7. The best formula is using 2 : 1. By Visualize Divisibility Theorem, x = 1, y = 2, n = 7, b = 10, (7, 2) = 1, (7, 10) = 1 and let m = 10p + q. Then 7 dividesp - 2q if and only if 7 divides m. Another convergence formula is using -5 : 1. By Visualize Divisibility Theorem, x = 1, y = - 5, n = 7, b = 10, (7, -5) = 1, (7, 10) = 1 and let m = 10p + q. Then 7 divides p + 5q if and only if 7 divides m.
3) A formula of 43 over base 300.
Table 2
The visualize array of ^3oo(43) = {1 : 1, ...}
2001 2002 2003 2004 2005 2006 2298 2299 3000
601 602 603 604 605 606 898 899 900
1001 1002 1003 1004 1005 1006 1298 1299 2000
301 302 303 304 305 306 598 599 600
1 2 3 4 5 6 298 299 1000
1 2 3 4 5 6 298 299 300
Note. Base 10 (gray), base 300 (white) and the best ratio (boundary).
We easily see that n = 43 is the postbase factor of b = 300, because 43 • 7 = 300 +1.
271 818 611 10710 = 33 167 106 237 007300.
By Postbase Factor Theorem, we use the different of alternate sum of digit, 33 -167 +106 - 237 + 7 = = -258 = -43 x 6.
So, 271 818 611 107 is divisible by 43.
4) Some formulae of 13 over base 1 000 000 and base 10.
Table 3
The visualize array of ,Rioooooo(13) = {-1 : 1, ...}
1'1 1'2 1'3 1'4 1'5 1'6 ... 1 1'999 998 1'999 999 2'0
1 2 3 4 5 6 999 998 999 999 |1'0
Note. Best ratio (boundary). We see that n = 13 is the prebase factor of b = = 1000000, because 13 • 76 923 = 1000000 -1.
Consider 3 937 376 385 699 289:0 = 3 937'376 385'699 2891000000.
By Prebase Factor Theorem, we use the sum of digit, 3 937 + 376 385 + 699 289 = 1'079 611.
1 + 79 611 = 79 612.
The best ratio is -4 : 1.
By Visualize Divisibility Theorem, x = 1, y = - 4, n = 13, b = 10, (13, 4) = 1, (13, 10) = 1 and let m =
= 10p + q.
If 13 divides p + 4q, then 13 divides m. So, 79612 ^ 7961 + 4(2) = 7969; ^ 796 + 4(9) = 832; ^ 83 + 4(2) = 91; ^ 9 + 4(1) = 13.
101 102 103 104 105 106 107
91 92 93 94 95 96 97 98 99 100
81 82 83 84 85 86 87 88 89 90
71 72 73 74 75 76 77 78 79 80
61 62 63 64 65 66 67 68 69 70
51 52 53 54 55 56 57 58 59 60
41 42 43 44 45 46 47 48 49 50
31 32 33 34 35 36 37 38 39 40
21 22 23 24 25 26 27 28 29 30
11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10
TpyAbi Brry Cepnn 3 № 1 2022
Table 4
The visualize array of Äio(13) = {-4 : 1, ...}
51 52 53 54 55 56 57 58 59 60
41 42 43 44 45 46 47 48 49 50
31 32 33 34 35 36 37 38 39 40
21 22 23 24 25 26 27 28 29 30
11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10
Note. Best ratio (boundary). 5) A formula for 24 combining 8 over base 1000 and 3 over base 10.
Since 24 is the product of prime number 3 and power of prime 8.
So the number which is divisible by 8 and 3 is also divisible by 24.
Consider the number 229986788520 is divisible by 24 or not. Since 3 is prebase factor of 10, adding all digits, the result 66 is divisible by 3. Since 8 is base factor of 1000, the last digit 520 is divisible by 8.
Therefore, 229986788520 is divisible by 24.
Conclusion. The above four theorems can formulate for many divisibility rules for any number over the any base. Some numbers are although difficult over base 10, they are easy over another base. Some numbers, such as primes, have direct rules, but some composites have combined rules. It is useful for all learners and teachers in mental calculating and in manipulate.
Список литературы
1. Davenpot H. The Higher Arithmetic. Cambridge: Cambridge University Press, 1999. 251 p.
2. Rina Zazkis. Divisibility: a problem solving approach through generalizing and specializing // Humanistic Mathematics Network Journal. Issue 21. 1999. Article 15. P. 34-38.
3. William Stein. Elementary number theory: primes, congruence, and secrets, a computational approach. Springer, 2009. 168 p.
References
1. Davenpot H. The Higher Arithmetic. Cambridge, Cambridge University Press, 1999. 251 p.
2. Zazkis Rina. Divisibility: a problem solving approach through generalizing and specializing. Humanistic Mathematics Network Journal, issue 21, 1999, article 15, pp. 34-38.
3. William Stein. Elementary number theory: primes, congruence, and secrets, a computational approach. Springer, 2009. 168 p.
Информация об авторе
Тиха Бо - доктор философии (математика). Университет Мандалая (Yangon-Mandalay Street, Nat Yay Kan Village, Amarapura Township Mandalay). E-mail: tbo290483@gmail.com
Information about the author
Thiha Bo - Doctor of Philosophy (Mathematics). Mandalar University (Y angon-Mandalay Street, Nat Yay Kan Village, Amarapura Township Mandalay). E-mail: tbo290483@gmail.com
Поступила после доработки 10.01.2022
