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Section 4. Mathematics
Section 4. Mathematics Секция 4. Математика
Drushinin Victor Vladimirinich National research nuclear University “MEPHI”, Sarov physico-technical Institute, Sarov, E-mail: vvdr@newmail.ru
The proof for hypothesis of Legendre — existence of a prime between two squares
Abstract: Analytically proved that there is at least one prime number between two squares of consecutive natural numbers. The proof is based on the two buffer zones on the classic sieve Eratosthenes and the application of the theorem about overlay of arithmetic progressions.
Key words: sieve of Eratosthenes, buffer zones, the amount of arithmetic progressions, Legendre conjecture.
Legendre’s conjecture was formulated by 200 years ago in 1808: on the interval (n2, (n +1)2) there is at least one prime number p, where n e N is the natural number. In the 25 volume encyclopedia “World of Mathematics" from 2014, the authors say that this hypothesis analytically so far not proven despite numerous attempts [1-3]. We have developed a method to create super grids of Eratosthenes, using the theorem of summation arithmetic progression (AP) and the application of probability theory. At the same time we first proved: an infinite number of pairs of primes-twins, primes par-fours, quartets of primes [4-6] and Euler- Goldbach conjecture [7]. This approach almost immediately proves the hypothesis of Legendre.
Write down a mathematical formulation of the classic sieve of Eratosthenes up to N. There is an overlay of infinite set AP has form s(pm, k) = pm + pm ■ k,k e N, on the numeric axis and removing these composite numbers. Begins the process with p1 = 2. When sequentially overlap of AP, the first not delete numbers (except number “1”) is a primes. The numbering of primes is in order { = 2 ;k2 = 3; p3 = 5; p4 = 7;...}. Ifyou take the first set (d +1) primes aJ+l ={p2;p3;p4 = 7;...;kakkd+1} ,you can select the two buffer zones on N. The first buffer zone A (d) = [ pd, p]+1 - 2] — interval on N. A characteristic feature of A (d) is that, if we removed all s (pm,k) where 1 < m < d in this interval, all of the remaining number are primes. Thus, the first not delete number pd not only gives prime, but also a whole set of primes on interval A (d).
The second buffer zone B (d) = [1, Td], where Td — the product of all primes from ad. Theorem
about1 overlay of AP s(pm,k) [5] says that the quantity, not deleted the AP numbers, on the B (d) exactly equal
d
to S (d ) = П(л -1) + d. There are probability detects not remote the number on B (d)
rn(d ) = 5 (d)/Td =(d / Td ) + П(1 -(1/Pk)). (1)
k=1
Immediately it should be noted that not all of the remaining number are prime numbers. For example, there is a “1”, which is not included in the system of primes. Composite numbers, which appear in the B (d)
, start with the pd+1 that does not remove a set ad. So as a [ Pd, p]+i - 2] contains not only remote primes then the probability of the number of primes N is
N (A (d ))=( - Pd - 1>{п(1 -(!/Pk )) + (d/Td). (2)
For example, when d=4, p4=7, p5=11, we have the A(4) = [7,119], BA) = [1,210], there are the number not removed numbers S (d) = 52. There are of them are not primes. This “1” and the five composite number are {121; 143; 169; 187; 209}. Length of the interval (the number of values in the buffer zone A(4)) there is a L(4) = 113, ®(4) = 0,2476, N(A(4)) = 27,9. The exact
number of primes are N ((4)) = 27.
Comparison of direct blending of AP and probability theory for large numbers always gives the actual number of deleted numbers not greater in magnitude than the calculation of probability theory, i. e. N (x2,xj )> N (x2,xj).
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Секция 4. Математика
Now there is key fact of evidence. Will prove a hypothesis of Legendre on the contrary. Let on the interval (n2, (n +1)) there is no primes. Hence, there is an inequality
Pd < n2 <(n +1)2 < pd+1. (3)
The number of all the numbers equal to 2n on the (n2, (n +1) ^.Therefore, the number of primes N on the theory of probability is
N (, (n +l) )= 2nj[n(l-(1/Pk))) + (/T )(4)
Since numerically conjecture of Legendre confirmed to n < 1010 then taking great n, you can ignore the second summand in the brace (4). Next we transform a(d) = П(-(!/Pk)).Since In(co(d)) = £n(1 -(1/pk)), we can use the following system of ratios
ln
\
1 --1
P J
\
f i Y
=-S 7
>-!
k j
k j
>- - .(5)
Pk-1 Pk
Using asymptotic equality of Legendre-Chebyshev type[5]
d i
Z“ = Ы(Сп ln (pd )) , (6)
pk
we find the number of primes is determined on the interval by the inequalities of the Legendre
(7)
By the condition the inequality (3) have -<Jpd < n, thus, inequality (6) can be strengthened
N((n +1)'))>)((,(n +1)’))>. (8)
With an increase of n and, consequently, increase of pd the right side of the inequality is a monotonically increasing, i. e. number of primes on the interval (pd, pd+1) seeks to infinity by the rule of L’Hopital. This contradiction proves assumption Legendre on the numeric axis.
The work reported at 8 All-Union conference “Mathematics and Mathematical Modeling” (Sarov, April, 2014) and 19 seminar ofyoung scientists in Nizhniy Novgorod (Nizhny Novgorod, May, 2014). The authors would like to thank prof., doctor ofphysico-mathematical sciences, head of RFYC-VNIIEF Y. N. Deryugina for discussion and support.
References:
1. Joaquin Navarro. Elusive ideas and timeless theorems. World ofmathematics, vol. 25, p. 58, Moscow De Fgostini, 2014.
2. Sizii S. V. Lectures on number theory, Moskow, 2007.
3. Dicson L. (E). History of the Theory of Numbers, v. II, CPC, NewYork, 1971.
4. Druzhinin VV, NTVP, 2014, №. 1, p. 22.
5. Drushinin V. V., Lazarev A. A., Sirotkina, A. G. Life Science Journal, 11, 2014 (10s), p. 346.
6. Druzhinin V. V., Lazarev A. A. NTVP, 2014, № 4, p. 21.
7. Druzhinin V. V. NTVP, 2014, №. 3, p. 14.
Shirokov Lev Vasilievich, Arzamas branch of the Lobachevsky State University of Nizhni Novgorod (UNN), candidate ofphysico-mathematical Sciences, associate Professor
E-mail: Shirokov1954@mail.ru
On radial spaces
Abstract: The article considers the class of radial space. We study the properties of the radial spaces associated with topological products.
Keywords: topological space, continuous mapping, compact, radial space, topological product.
Широков Лев Васильевич, Арзамасский филиал ННГУ им. Н. И. Лобачевского
кандидат физико-математических наук, доцент E-mail: Shirokov1954@mail.ru
О радиальных пространствах
Аннотация: В статье рассматривается класс радиальных пространств. Изучаются свойства радиальных пространств, связанные с топологическими произведениями.
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