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Section 4. Mathematics
Section 4. Mathematics
Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich, National research nuclear University "MEPHI” Sarov physical-technical Institute Department of mathematics, Sarov E-mail: vvdr@newmail.ru
The extension of euler-fermat’s theorem and the rationale of carmichael numbers
Abstract: It was carried out the generalization of the theorem of Euler, which includes the small Fermat’s theorem, for the case of different degrees of base, if it is mutually simple with arbitrary integer. The matrix is composed of deductions and specified its properties of symmetry. It is given the applications of new ratios.
Keywords: Euler’s theorem, Fermat’s little theorem, the theory of comparisons, residues, deductions.
In comparison theory Euler’s theorem (TE) has the form ak = l (mod m) ,where, in the general case, the composite whole integer m > 1, base ak and m are coprime, i. e. (ak, m ) = 1 and p = ty(m ) — Euler function. и is the number of numbers ak less than m . If
n
there is the canonical decomposition m = Пpa , then
П k =1
fd = The -' ■(pk - '),where pk — here and further
к='
primes p e P. и — even number. For instance, (p(20) = p = 8, ak =7, 78 = 1 + 288240• 20 [l-3]. A special case of the TE, when m = p, is Fermat’s Little Theorem ap 1 s l(modp).A generalization of Euler’s theorem and Fermat’s theorem was given in articles Dru-zhinin [4, 5] in the form of a theorem FED:
If (ak ,m) = 1, 0 < n < p, then for any ak necessary and sufficient comparison
ak" = dn (modm), (1)
where dn are from comparison ankdn = l(modm).
Proof. Let n = 1. Then ak 1 s d1 (modm).By multiplying the comparison by ak, we get
ak = ak ‘ d1 (modm) . There is the hypothesis ak ■ d1 = l(modm).Therefore ak = l(modm),and we get the TE. In the case of n = 2, we multiply comparison ak 2 s d2 (mod m) by ak and again obtain TE. The theorem is proved.
One consequence of TE is Fermat’s Little Theorem, which is necessary and sufficient in correct the formulation.
If for any 2 ^ a < (s -1) a comparison is made as 1 s l (mod p), it is a necessary and sufficient condition that s been a prime number. In the symbolism of Fermat’s Little Theorem in our edition looks like V ak 2 < pk <(s p l) a5-1 s l(mods)o s e P. (2)
The requirement — for any 2 <ak <(s-1) - create sufficiency of Fermat’s little theorem. When there was not it, then there is a composite pseudo-primes S and the numbers of Carmichael C, satisfying the comparisons: aS 1 s l (mod P) on one ground a and aC 1 s l (mod C) for all bases ak are coprime to C. For example, S = 341 = 11 • 31 included in this comparison only for base a = 2 and for the base a = 3 deduction is notequalto «1». Carmichaelnumber C = 561 = 3 -11 -17 enters the fomula Fermat’s little theorem on all bases a, except a = {3k; 11k ;17k }.
Thus, the pseudo-primes and Carmichael numbers should be considered from the point of view of TE, but not Fermat’s little theorem. In this case, there are no contradictions.
When calculating matrix FED deductions theorem of FED we will use the full system of residues in the form of remainder of the division 0 < dn < m — 2, and the last remainder (m -1) is replaced by the residue of «-1». Calculate d1 in the example m = 20, /4 = 8, a = 7. Solution of comparing 7• d1 = l(mod20) gives dx = 3. Wherein 77 = 3 + 41177 • 20, i. e. FED generalization is performed.
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The extension of euler-fermat’s theorem and the rationale of carmichael numbers
To calculate dn, or more strictly dn (m; ak ),, we introduce an interim coefficient sn by rule
an = sn (modm), which gives the equation sn ■ dn = 1 (nodm). Sets {ak},{xn} and {dn} consist of y number less than m and coprime to m, but are arranged in a different order. They always have a «1» and (m -1). (m -1) replaced in the matrix deductions for {s„} and {dn} to «-1». In the considered higher in example ak {1;3;7;9;11;13;17;19} .If in (1) n = 0, we obtain the TE, i. e. d0 = 1. Next, we will consider the ligaments (sn \dn), among which there are two trivial
(l|l) and (—1| — 1) .The number of non-trivial ligaments without permutations sn and dn is all (u/2)-l) . There are recurrent comparisons facilitating calculation of ligaments: sn-1 ■ a = sn (modm) and
dn ■ a‘k = dn-t (mod m) . Table № 1 is a matrix deductions for m = 20.
We single out the following properties of the matrix ligaments. 1. In the ligaments numbers can be permuted, and both numbers include only one ligament. 2. The left column (a1 = 1) and the first line n = 0 (in TE) consist of unit ligaments (1|1).
Table № 1. - Ligaments (n\dn) for m = 20. к — numbers the base к .
k 1 2 3 4 5 6 7 8
a n 1 3 7 9 11 13 17 19
0 (11) (11) (11) (11) (11) (11) (11) (11)
1 (11) (37) №) (9|9) (urn) (1317) (1713) (-11-1)
2 (11) (9|9) (9|9) (11) (11) (9|9) (9|9) (11)
3 (11) №) (37) (9|9) MO (1713) (1317) (-11-1)
4 (11) (11) (11) (11) (11) (11) (11) (11)
5 (11) (37) №) (9|9) M0 (1317) (1713) (-11-1)
6 (11) (9|9) (9|9) (11) (11) (9|9) (9|9) (11)
7 (11) № (37) (9|9) M0 (1713) (1317) (-11-1)
3. In the matrix, there is one central row with n = в = (ц / 2) , which consists of ligaments (1|1) or (—1| — 1) .For large m these lines, consisting only of the ligament (1|1) or (—1|-1) maybe a few. We will call them as central. Often, as in this example, the entire central line consists of ligaments (1|1). In table № 1, в = 4 , and this line is selected. It is the horizontal axis of symmetry for each column sp+k = dp-k and sp-k = dp+k .
4. There is such a modification TE aи/2 = (±)l(modm ). The sign (±) indicates that one of the sign is realized.
5. If dp (a ) = 1, that dn (a,) = dp+n (a,),0 < n <(p-1).
If de(ak ) = -1, that dn (ak) + dp+n (ak ) = m
0 < n <(P -1). 6. There is a vertical axis separating columns P = ^/2 and (в + l) = (/r + 2)/2 . Table № 1 it is the columns a4 = 9 and a5 = 11. The vertical axis of symmetry highlighted dotted line. 7. For the even lines n there is the mirror image ligaments about a vertical axis. If n is odd, then there is the anti-mirror reflection.
d2t {ap-k ) = d2t {ap+\+k ) • d2t+1 {ap-k ) + d2t+1 {ap+l+k ) = m-
8. Always comparison is performed (m — 1) =
= (—1)” (modm e 9. |(a (a ( -l)/(a - l/-/dn |: m.
10. In line with n = 1, all d1 are different and run through a value from «1» to «—1», i. e. (m — 1).
Thus, for the construction of the residues matrix for theorem FED is sufficient to know only the upper left block of the matrix: the columns 2 < к < fi and rows 1 < n < в .
From the perspective of our theorem it is clear the origin of the pseudo-primes and Carmichael numbers. The pseudo-prime number is composite number of S, satisfying Fermat’s little theorem on one base. Explain why with S = 341 = 11-31 the 2341 1 s \ (nod341) . This is not surprising, since ^ = 10 • 30 = 300 and 2340 = 240 = d260 (341;2)(mod 34l).Thusd260 ( 341;2 ) = 1. For other bases, for example, a = 3 Fermat’s Little Theorem is not fair, 3340 = 56 (nod 341).
As for the number ofCarmichael C = 561 = 3 -11 -17, then its ^ = 2 • 10 • 16 = 320 . According to the above
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Section 4. Mathematics
algorithm, we calculated the matrix of ligaments 320 x 320 for this number and found four central lines of (l|l) for n = 0;80; 160;240 , consisting only of the ligaments (1|1) . Therefore a561-1 = a560 e a240 e
= d80 (561;a )(mod561) si (mod 56! ).The next number Carmichael C = 1105 = 5 • 13 • 17 has ^ = 768 . Matrix deductions has the 16 central lines from (1|1), i. e. atSt = l(mod 1105). Because й^04 = af23 =
= l(mod 1105) .The same is true for other numbers Carmichael. Our approach allows us to generate a new num-
ber C and indicates their infinite amount. This odd number C, composed of at least three multipliers, and have the form C = 4k +1.
Th e result of the FED s theoremare new formulas for primes. If p = 8k +1, then there is the congruence
42k-1 = (6k + l)(mod (8k +1)). (3)
If p = 8k — 1, there is the congruence
44k-1 = l(od (8k -1)). (4)
Comparisons (3, 4) is a modification of Fermat’s little theorem on the necessary features.
References:
1. Sizii S. V. Lectures on the theory of numbers, M. FIZMATLIT, 2007.
2. Sushkevich A. K. Theory of numbers. Kharkov, KSU, 1956.
3. Ingham A. E. The distribution of primes. M. Com. Book, 2007.
4. Druzhinin V. V. NTVP, № 5, 28-29, 2015.
5. Druzhinin V. V. NTVP, № 5, 30-32, 2015.
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