CC BY
32
Section 5. Mathematics
Section 5. Mathematics
Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich, Smagin I. R., Hromov N. O.
National research nuclear University “MEPHI” Sarov physical-technical Institute Department of mathematics
E-mail: [email protected]
Summary carmichael numbers
Abstract: On the basis of the generalized Fermat-Euler’s theorem it was found an algorithm for obtaining the numbers of Carmichael, and new options for comparison deduction to unity. It was proposed new necessary conditions for the existence of primes.
Keywords: Euler’s theorem, Fermat’s little theorem, the theory of comparisons, deductions, Carmichael number.
When about 400 years ago in 1640 Fermat formulated his little theorem: ap 1 = 1 (modp),where p is a prime number, (a,p) = 1 — mutual easy numbers, it was seemed that a simple analytical formula received generation ofprimes. But in 180 years, it was discovered the composite pseudo-prime number 341 = 11- 31, which (only) satisfy Fermat 's small theorem when a = 2. In 1910 it was found the number C = 561 = 3 • 11-17, which give comparison a5611 = 1 (nod56l) for all a : (a,56l) = 1. Thus, in old formulation Fermat 's small theorem become necessary but insufficient. Another formulation this theorem allows and her sufficient: if for all a coprime for n an 1 = l(modn)
, then this required and sufficient condition ofprime n. But enumeration of all basics gives non competitive finding of primes compared to the method of Eratosthenes [1-3].
At present a lot of Carmichael numbers and it is known, that the number of their is infinite. However, the nature of their appearance and how to work around these pitfalls may be supplemented. In our article the issue resolved on the basis of generalized Euler-Fermat theorem FED [4-6].
Euler’s theorem (ET) has the form aß = l(rnodm) , where the integer m > 1, (a,m) = 1 and p = ty(m) — the number of num-
bers less than m and coprime to m. If m = Upa, then
p = -1 (Pt -1)), where pk — primes that are included in the
canonical decomposition m. p — even number. A special case of the ET when m = p, is Fermat’s Little Theorem. The generalization of Fermat’s little theorem and ET is given in [3]. It is formulated in the following theorem. If (a,m) = 1, 0< n < (p — l),then
a= dn (modm), (1)
where dn is found out of the equation {a" -dn) = 1{modm).
To calculate dn or more strictly dn (m;ak), we introduce an intermediate coefficient sn according to the rule of indices a" = sn (modm) which gives equation (sn • dn) = 1(modm) .The sets {ak},{sb} and {dn} consist of p numbers less than m and coprime with m. They always have a1 = 1 and = (m — 1). Next,
we will consider bundles (sn\dn), among which there are two trivial (ill) and (m - 11m -1) = (-11 — l) .Non-trivial bundles without considering the permutations of sn and dn are (p— 2)/2).There are recurrent comparisons that facilitate the calculation of bundles: (_1 • a ) = sn (modm) and (dn -a' ')=dn_t (modm) .In table. No. 1 it is showed a matrix FED of bundles for m = 20.
Table 1. - The matrix is FED for m = 20, p = 8. k numbers the base a.
k 1 2 3 4 5 6 7 8
\ a n 1 3 7 9 11 13 17 19
0 (111) (111) (111) (111) (111) (111) (111) (111)
1 (111) (3|7) (7|3) (9|9) (1111) (13117) (17113) (-11-1)
2 (111) (9|9) (9|9) (111) (111) (9|9) (9|9) (111)
3 (111) (7ß) (3|7) (9|9) (1111) (17113) (13117 ) (-11-1)
4 (111) (111) (111) (111) (111) (111) (111) (111)
5 (111) (3|7) (7|3) (9|9) (1111) (13117) (17113) (-11-1)
6 (111) (9|9) (9|9) (111) (111) (9|9) (9|9) (111)
7 (111) (7ß) (3|7) (9|9) (1111) (17113) (13117 ) (-11-1)
Table No. 1 Central line with n = ß = 4 consist of {(ill)} .We ing of the remnants of the division with the replacement of the redenote it as ß(+). We use the main system of deductions, consist- mainder (m -1) to deduct (-1). Symmetry properties of the matrix
18
Summary carmichael numbers
described in [4-6]. A special role in the occurrence of Carmichael numbers has a central row of the matrix at n = p /2 = ß . It always consists of bundles (ill) or set of {(ill);(—11 — l)} . This so-called Legendre symbol. As can be seen from the table. No. 1 to build the whole matrix is sufficient to know only the bundles of the upper-left block: 1 < k < ß;0 < n <ß . Properties of symmetry the line.
sn,ß-k+1 = sn,k ; d„,ß-l+1 = dnk , if n is even. sn,ß-k+1 = m - snk , if n is odd.
Symmetry in the next column. If s ß,k = dßk = 1, then
Sß+nk = Snk ; dß+nk = dnk . If Sßk = dßk =-1 , then Sß+nk = m - Snk ;
dß+nk = m - dnk .
For example (see tab. No. 2) will give the FED matrix for m = p = 13, p = 12, ß = 6.
Table - 2. Ligament (\dn) for m = 13. k numbers the base a.
k 1 2 3 4 5 6
\ a n 1 2 3 4 5 6
0 (111) (111) (111) (111) (111) (111)
1 (111) (2\7) (3|9) (410 ) (5|8) (6|11)
2 (111) (410) (9B) (3|9) (-11-1) (1014 )
3 (111) (8Б) (1111) (-11-1) (8Б) (8Б)
4 (111) (3|9) (3|9) (9ß) (111) (9ß)
5 (111) (6|11) (9ß) (1014 ) (5|8) (2\7)
6 (111) (-11-1) (111) (111) (-11-1) (-11-1)
In table. No. 2 central line with n = ß(±) = 6 consists of
H"1)}. , . , .
View ofthe central line ß(±) or ß(+) plays acrucialrole inthe appearance of Carmichael numbers, they occur only when ß (+). In the FED matrix can occur a single row at n = ßk = p / s2 , also consisting only of (ill). Ifwe take the line with the smallest such n = ßt, the matrix will be permeated with the lines with (ill), separated from each other on ßt. Hence there is the condition for the existence of Carmichael numbers: (C — 1)-ßt. Example 1. C = 561 = 3-11 -17, p = 2 -10-16 = 320, ß = 160.Inthe matrix 320x320 there are four single rows when n = 0;80;160;240. So ak“ = 1 (mod56l), even if
we go by the column upward outside of the matrix. When e = 7, the exponent 560 = 561 — 1. Example 2. C = 1100 = 0-13 • i 7, p = 768, ß = 384. In the matrix of768x768 there are sixteen single strings when n = 0;48;96;.. ,;720. So a148 = 1 (mod 1105) .If e = 23, exponent 1104 = 1105 — 1. Example 3. It was initially thought that Carmichael numbers are of the form C = 4t +1. But subsequently it were found the number of forms C = 4t — 1. One of these numbers 8911 = 7-19-67,,u =7128, Д =891, 10-891 = 8911-1 . Analysis of Carmichael numbers showed that C -1 = ep / (f ■ 2l) .In table. No. 3 we show the calculated ratio between the parameters of some Carmichael numbers.
Table - 3. The parameters of Carmichael numbers
C M t; ß, Form of С
1729=7-13-19 1296 12; 108 C -1 = 16p/12
2465=5-17•29 1792 8; 224 C -1 = 11p/8
2821 = 7-13 • 31 2160 36; 60 C -1 = 47p/36
6601 = 7 • 23 • 41 5260 4; 1315 C-1=5p/4
10585=5•29•83 8064 16; 504 C -1 = 21p/16
15841=7•31•73 12960 36; 360 C -1 = 44p / 36
Thus, a necessary condition for the existence of Carmichael numbers is a single line with n = ß . Under what m is this happening? If m = p, then the central row consists of j(lll)ииз(—11 — 1)},
i. e. these m do not give the number of Carmichael. In our analysis of the single central row occurs for m =2" и m = p2 • p2 ,when pj > 3, p2 > pj and some other combinations.
Will discuss the extension of the notion of Carmichael number, namely the comparison of the form aD - = 1 (mod D). First take t = 0. Comparison of form aD = l(modD) occur even for D = eßt. Examples: D = 6, p = 2; D = 8, p = 4; D = 12, p = 4; D = 16, p = 8 . In
the General case (2t +1) = 1 (mod 2" ).Next, consider the number D = 3 • p, p = 2(p — l), ß(+) = (p — l),p + ß = m — 3 . This comparison gives a3p 3 = 1 (mod 3p) . If there are two primes and p = kp1 — 1, then there is the correct comparison pp "1 = k (mod p). Example. 53 = 2 • 33 — 1, 349 = 2 (mod53)Thenumber D = 15 is the generalized Carmichael numbers of the form aD+1 = l(modD) ,
since p = 8, and 2p = 16. The generalized number of Carmichael = 24 gives a comparison a = a = a = a = lymoaD) since in the FED matrix for this number p = 8 and the string when n = 0;2;4;6 are composed of bundles (ill).
Thus, the properties of the FED matrix allow to find, apparently, any comparison ofthe form adD ' = l(modD) and t = 1, i. e. the real numbers Carmichael is a special case.
Using the properties of FED matrixes, you get a new the necessary attributes of a prime. Here are some of them: if p = 8k ± 1, then 2(p-1)/2 = l(modp); if d = 12fc ± 1, then 3(p-1)/2 = l(modp);; if p = 10k± 1, then 5(p-1)/2 = l(modp); 2p-2 =((p +1)/2)modp) (p - 2 )2=(p -1)/2 )modp); if p = 3k +1, then
3p-2 =(2fc + l)modp) and (p-3) = к(modp).
The authors thank prof. Shevyahova N. S. for a discussion of the results.
19
Section 5. Mathematics
References:
1. Sizykh S. V. Lectures on the theory of numbers, M. FIZMATLIT, 2007.
2. Vinogradov. I. M. Fundamentals of the theory of numbers, Nauka, Moscow, 1981.
3. Song Y. Yan. Number theory for computing, Springer, Stanford, 2001.
4. Druzhinin V. V. NTVP, No. 5, 2015, 28-29.
5. Druzhinin V. V. NTVP, No. 5, 2015, 30-32.
6. Druzhinin V. V., Lazarev A. A. Austrian journal of technical and natural Sciences, № 9-10, 2015, 18-20.
Khubaev Georgy N., doctor of economic Sciences, professor of Rostov state University of Economics (RINH) E-mail: [email protected] Scherbakov Sergey М. doctor of economic Sciences, professor of Rostov state University of Economics (RINH) Shirobokova Svetlana N. candidate of economic Sciences, associate Professor, Platov South-Russian State Polytechnic University (NPI)
Conversion of idef3 models into UML-diagrams for the simulation in the sim system-UML
Abstract: The problem of conversion of IDEF3 models of the business processes in UML diagrams for system simulation SIM UML. The conceptual approach and features of the mechanism of transformation of individual elements of the IDEF3 model and software implementation of the algorithm conversion are described. The conversion results enable to use previously created IDEF3 model for the synthesis of simulation models and subsequent optimization of resource business processes.
Keywords: conversion IDEF3-models, system simulation SIM-UML, the synthesis simulation models, optimization of resource intensity of business processes
Introduction. The authors in [1-4] provided a conceptual idea of Automated IDEF0 models converter into UML-diagrams, which further was reflected in world primarily developed Converter «ToADConverter» [5]. However, the issue of converting of IDEF0 standard diagrams, as well as the diagrams of IDEF3, DFD types into UML-diagrams for ensuring automated synthesis of simulation model of business process (for example, with the use of the Constructor [6]), is still important. As in this case, the efforts for the developing of simulation model of any business process are reduced in dozens of times. The BPWin file format used previously for IDL export enabled to export the diagrams of decomposition in IDEF0 standard only. However, the models of the real processes are frequently completed by DFD and IDEF3 diagrams. Such mixed model, three aspects, enables to use and automatically approve the most popular notes of business-processes simulation, provides complex description of object region. Lately appeared pack AllFusion Process Modeler 7 enables to export mixed models in XML format efficiently. With this information about all diagrams (IDEF0, IDEF3, DFD) is preserved. In such case, the problem of syntaxes analysis of XML-file with saved mixed model becomes important.
It is known, that IDEF3 standard is a methodology of process description, which considers the consequence of their execution and causality between the situations for the structural presentation of knowledge about the system, description of objects' conditions changes.
Visual simulation in IDEF3 standard is one of the most popular means of geographical representation of business processes.
Consequently SUM system-UML [6], developed within the implementation of process-statistical approach to the calculation of expenditures [7], enables, basing on UML-diagrams ofbusiness processes (precedents diagram and activity diagram), synthesize simulation model of the process. Basing on the results of simulation modelling, the expenditure costs are evaluated, and the most resource intensive processes and operations are defined. The experiments with the model provide possibility for the reduction of resource intensiveness of the studied subsetting of business processes [7-10].
Converting of IDEF3-models into UML-diagram provides possibility to optimize the processes' resource intensiveness in different industries. Thus, the resource expenditures for citizens' access to particular services (vehicles registration, enterprise opening, passport exchange and etc.) in different countries can be compared. Having built visual and imitation models the expenditures of labor and other resources can be quantitatively compared, for instance, social servants, commercial enterprises' employees, individual entrepreneurs and citizens for the service implementation provided by the state and municipal management authorities.
Features of the Proposed Converting Algorithm. Converting of IDEF3-models into UML-diagrams is performed with the orientation on CASE-mean for the simulation of business-processes Allfusion Process Modeler, which enables to create the diagrams in notations IDEF0, IDEF3 and DFD.
On the first stage, the syntax analysis (parsing) ofxml-file models is provided to gain access to model's units (activity, junction) and junction arrows, to the features and characteristics of model elements.
20
