Sciences new aspects of euler''s theorem from the theory of comparisions Текст научной статьи по специальности «Биологические науки»

MATHEMATICAL SCIENCES

NEW ASPECTS OF EULER'S THEOREM FROM THE THEORY OF COMPARISIONS

Druzhinin V.

Doctor of phys-matematical Sciences, Professor, the head of the Department of higher mathematics

Sirotkina A. the head of the Institute SarFTI NRNU MEPhI, Sarov

Abstract. A generalization of Euler's theorem from the theory of comparisons done in case of arbitrary values of degrees. The matrix deductions received and determined its properties. Application of the formulas given.

Keywords: Prime numbers, Euler's (p -function and theorem, comparisons, deductions.

Euler's theorem (TE) was formulated nearly three hundred years ago in 1732, but still is modified and gives new results. Euler was the first who proved small Fermat's theorem. Itself the small Farm's theorem was formulated nearly a hundred years ago in 1640. For the formulation of TE need to find the factor n of a composite positive integer m = n7k=1pik. № = <p(m) =

Wk=1(Pkk-1 • (Pk - 1)), where p(m) is famous <p-function of Euler, which gives the number of elements ak less than of m and are coprime with m. It is written as ak Lm (or (ak ,m) = 1),1 < k < n. We note some properties of the sets {ak}. 1. n - always is an even number. 2. In {{ak} always includes numbers «1» and (m — 1). 3. The point of symmetry is in the middle of the set {ak}, which gives ak = (m/2) ± s. When m is even, then s are integers, with m is odd, then s are palusalue number. In the future, we have this point of symmetry is needed. p k- primes (PN), numbered in ascending order: p1 = 2; p2 = 3; p3 = 5;... . In the symbolism of TE has the form [1, 2]

(1)

^ = 1(mod m),

lk

remainder «1». For example, = 22 • 5, ^ = 8, the set is {ak } = {1; 3; 7; 9; 11; 13; 17; 19}, 118 = 1 + 10717944 • 20. If m is the PN, then TE goes to the

i. e. when you divide ak on m there is always a

small Fermat's theorem. In this article, when aim, we formulate a generalization of TE in the form

an = e(a; m; n) + r(a; m;n) •m = e(a;m;n)(mod m), (2)

i.e. we allow any degree n = 0;1; 2; 3;...; n; n + 1;... . In textbooks and monographs we have not found this material. In (2) e(a; m; n) is the smallest to modulo m deduction (SMD), —(m/2) <e< (m/2). We will first use

1 < a < (m — 1) and n < p. The General formula for the subtraction of e has the form

a • e(a; m;n — 1) = e(a; m; n)(mod m), (3) i. e., moving along the column from top to bottom, we find all deductions for a given a. Immediately write down the obvious properties of SMD (here in after t = 0; 1; 2; 3;..):

e(a; m; 0) = 1; e(a < m/2 ; m; 1)

= a; e(a > m/2 ; p; 1) = a — m; e(a; m;n + tp) = e(a; m; n); e(1; m; n) = 1; e(m — 1; m; n) = (—1)n; e(a; m; p/2)

= ±1; e(tm;m; 0) = 1; e(tm; m;n> 0) = 0; e(a + tm; m; n) = e(a;m;n). (4)

As an example, we write the set e(a; m; n) in the matrix of the Euler-D for m = 15, ^ = 8 (see table1).

Table1

The matrix e(a;m; n) of the Euler-D SMD for m = 15

n\a 1 2 4 7 8 11 13 14 15

0 1 1 1 1 1 1 1 1 1

1 1 2 4 7 -7 -4 -2 -1 0

2 1 4 1 4 4 1 4 1 0

3 1 -7 4 -2 2 -4 7 -1 0

4 1 1 1 1 1 1 1 1 0

5 1 2 4 7 -7 -4 -2 -1 0

6 1 4 1 4 4 1 4 1 0

7 1 -7 4 -2 2 -4 7 -1 0

8 1 1 1 1 1 1 1 1 1

In the matrix stands for vertical (bold) line of symmetry separating the columns at the point of symmetry of the set {ak}. Among the numbers e(a; m; n) exists the recurrence relation horizontally e((m/2) — s; m; n) = (-1)ne((m/2) + s; m; n). (5) We observed vertical transfer units in n = {0;1;2;...;//} from the top down e(a; m;n + p/2) = e(a; m; y./2)e(a; m;n < p/2). (6) These properties are similar to properties of the deduction matrix of Farm-D [3]. From (5-6) we see that the first p./2 numbers in the row when n is even experiencing a mirror reflection about the vertical axis of symmetry, and if n is odd that is mirror reflected with change of sign. In the column of the first n/2 SMD from the top down again in the next n/2 columns, if

(a,m,^/2) = 1 , and they are repeated with a change of sign if e(a,m,^/2) = —1 . Central line with n = H/2 (highlighted in the table 1) always consists of «±1», as well as in the matrix Fermai-D [3]. In the matrix may be a few lines consisting of «1». The bottom line of the table 1 with n = n is itself TE. As we can see, to construct the set of all deductions is enough to know the deductions of the left upper rectangle: 2 < a < m/2, i. e., (p/2) numbers horizontally and 2 < n < n/2 numbers vertically. The first two rows are standard in all matrices. Here are the basic rectangles for the other three m, see table 2. We see the matrix with m = 24 three single line with e = 1 for n = {2; 4; 6}.

Table 2

The basic rectangles of the matrices of the Euler-D for m,^ = {10,4; 21,12; 24,8}

n\a 1 2 4 5 8 10

2 1 4 -5 4 1 -5

3 1 8 1 -1 8 -8

4 1 -5 4 -5 1 4

5 1 -10 -5 -4 8 -2

6 1 1 1 1 1 1

24

21

10

n\a 1 5 7 11

2 1 1 1 1

3 1 5 7 11

4 1 1 1 1

n\a 1 3

2 1 -

Discuss some applications of matrices Euler-D.

1. If we know SMD, i.e. e, we can easily find remnants of division 0 < c < m when is dividing an on m, an = c + D • m. If 0 < e < m/2, then e = c, if — (m/2) < e < 0,then c = m + e.

2. Occurs quick ways of dividing integers. This interesting aspect of the use of the generalized theorem of Fermat - Euler and related matrices and gives to simplify arithmetic operations. In computer calculations, as is known, the basis are the operations of addition and subtraction. Multiplication is repeated addition occurs relatively quickly. The most time consuming area of division when the dividend on the private numbers from left to right. Meanwhile it is known that if the dividend is divisible by the divisor, then divide the number you can from his end, i.e. from right to left. While searching for a number in private is not necessary, as in the first case. Each digit in private is just the multiplication table. For example, it is necessary to divide «221» obviously a multiple of the number «13» is the number «13». In order from «3» to «1» need of «3» multiplied by «7», there are no other options. Multiply 13^7 = 91. After you subtract we have 221 — 91 = 130. Next, divide the «13» to «13» and get a result 221/13 = 17. Thus, the division comes down to the rapid multiplication and subtraction. This in principle can speed up the

fission process and reduce the time of calculations. The matrix of the Euler-D just gives you a deduction when dividing two numbers, and if this deduction be removed from the dividend, then you can divide the area, quickly from right to left.

3. Since r(a;m;n) in (2) is an integer, as (an — e(a; m; n)) a multiple of m, it is possible to divide this bracket specified above. For numbers r(a;m;n) revealing new properties. Obvious: r(1; m; 0) = 0; r(tm < a < tm + (^/2); m; n) = t; r(tm + (ji/2) <a<(t + 1)m; m;n) = t + 1. Non-obvious property has views

r(a; m;n) = a • r(a; m;n — 1) + S(a; m; n). (7) A corrector 5 gives the possibility to calculate the coefficients r bottom row factor r in this column. Thus S(a; m; n) have the following interesting properties: S(a; m; n) = [a^ e(a; m;n — 1) — e(a; m; n)]/m; (8) S(m — 1; m; n) = (—1)n+1; 5(1; m; n) = 0;

S(a;m;n + (tp)/2) = S(a;m;n), (9)

i. e. row of the matrix of the Euler-D the parameter 5 are periodic in repeats. We give the sets r(a; 15; n)and S(a; 15; n) (see table 3) from the top down through ^/2=4 rows. The table allows to calculate r(a;15;n)at the values r(a;15;n — 1) and 5(a; 15; n). For example,

Tablel

The matrix (r ; S) for m = 15_

n\a 2 4 7 8 11 13 14

1 0; 0 0; 0 0; 0 1; 1 1; 1 1; 1 1; 1

2 0; 0 1; 1 3; 3 4; -4 8; -3 11; -2 13; -1

3 1; 1 4; 0 23; 2 34; 2 89; 1 146; 3 183; 1

4 1; -1 17; 1 160; -1 273; 1 976; -3 1904; 6 2561; -1

5 2; 0 68; 0 1120; 0 2185; 1 10737;1 24753; 1 35855; 1

6 4; 0 273; 1 7843; 3 17476; -4 118104; -3 321787; -2 501969; -1

7 9; 0 1092;0 54903;0 139810; 2 1299145;1 4183234;3 7027567;1

8 17; -1 4369; 1 384320; -1 1118481;1 14290592; -3 54382048;6 98385937; -1

We see from table 3 that the numbers 8 ( cost after (;) in the column) are repeated from the top down through n/2 = 4 rows. The table allows to calculate H/2 = 4 at the values r(a; 15; n- 1) and S(a; 15; n). For example, r(8; 15; 4) =

= 273; r(8; 15; 3) = 34; 5(8; 15; 4) = 1. So 34^8 + 1 = 273. On the other hand, ((84 - 1)/15) = 273.

This property is preserved if we extend the table 3 to the left and down to any size. For example, we find r(7; 15; 9) = 7 • 384320 = 2690240. Table 3 allows you to find all r(a; m; n) from «1»from the top down with small payments according to the formula r(a; m; n) = r(a; m; 1)an-1 + S(a; m; k) an-k. (10)

4. Consider the great Fermat's theorem. The amount of the deductions in line to confirm this hypothesis. Let m = 15, see table 1. Any combination of numbers in the degree of «4» gives x4 + y4 — z4 ± 0, if {x; y;z} 115 . Because in the row with n = 4 are the number «1», a combination of deductions can not give equality to zero.

5. Becomes clear the mystery of the composite numbers Carmichael. The first is the number found in 1910, K1 = 561 = 3^11^17 , unexpectedly satisfy Fermat's theorem for all bases a 1K1 aKl-1 = 1(mod 561). <p(K1) = n = 2^10^16 = 320. Central line with n = yi/2 = 160 consists of «1». The matrix of the Euler-D has and other single line with a period of "80", i.e. the n = {0; 80; 160; 240; 320; 400; 480; 560; 640;...}. It gives a561-1 = 1(mod 561). The other Carmichael number K2 = 1729 = 7 • 13 • 19,^ = 1296 in the matrix of the Euler-D between the individual rows of «108». Since 16 • 108 = 1728, then for all a 1 1729 is the place a1729-1 = 1(mod 1729). On this basis, we obtain the algorithm for generating numbers Carmichael and the ability to use small Fer-mat's theorem to generate a PN.

6. The matrix of Farm-D gives a simple formula for the exact number of PN on the stretch Ln = [1, Tn],

where Tn = ££=2 pk. Within this segment there are three groups of PN. An = {3; 5; 7;...; pn} - these small PN create Tn. The second group Bm = {pn+1 = Pi; Pn+2 = P2; .■■; Pn+m = Pm} - m additional medium of PN, where pm < JT^ < pm+i. Cm =

{Ps(i); Ps(2);.; Ps(t);.; pS(m)} is large gr°up of large PN that are from the inequality pt • ps(t) <Tn < pt • Ps(t)+1 . Here s(t) - the number of the drive in a General manner. The exact number of PN on the stretch of Ln is defined by the formula

n(Ln) =

ф(Тп) + 2(т+1)(п-1)

2

Y.?=i(s(t) -1).

(11)

Example. n = 5; T5 = 3 • 5 ^7 • H = 1155; B6(t) = (13^; 172; 193; 234; 29s; 316); C6 =

{p23(l) = 83; Pl9(2) = 67; Pl7(3) = 59; Pl5(4) =

47; pi2(5) = 37;pi2(6) = 37}. <p(1155) =\i = 480; ^(1155) = 268 -(22 + 17 + 14+ 11 + 7 + 6) = 191. Thus, over the interval from «1» to «1153» is exactly «191» PN. The empirical formula of Legen-dre n(N) = N/((lnN)- 1.08366) [1,2] дает «193.5» ПЧ.

This method also allows you to find the greatest PN in this segment. The number «1153» does not consist of PN from A5 by definition. Check just multiplication «29 • 37», as it is one ends in «3». But 29 • 37 = 1073. Therefore, «1153» is PN.

7. Generalized Euler's theorem can also participate in solving algebraic and linear Diophantine equations.

REFERENCES:

1. Graham, Z., Knuth, D., Patashnik O. Concrete mathematics, Moscow, MIR, p. 322, 1998 .

2. Buhshtab V. V. The theory of numbers. M. S.-P. Lan. 2015.

3. Druzhinin V. // NJDIS, No. 13, p. 43-45, 2017.