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Proof of euler-goldbach’s conjecture
Section 4. Mathematics
Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich National Research Nuclear University "Moscow Engineering Physics Institute" Sarov Physico-Technical Institute, E-mail: vvdr@newmail.ru
Proof of euler-goldbach’s conjecture
Abstract: We analytically proved that every even number greater than “4” can be represented as the sum of two primes. We give a simple algorithm to determine the number of such pairs and their particular view on any even number. A formula is derived for the approximate number of pairs of Euler-Goldbach for arbitrary even number. Calculations are confirmed the created algorithm.
Keywords: Euler-Goldbach’s conjecture, the sieve of Eratosthenes, the number of pairs of the Euler-Goldbach.
In 1742 in a letter to Euler Goldbach suggested that every odd number is the sum of three primes. This ternary Goldbach’s conjecture was proved by Academician Vinogradov in 1937 for prime numbers greater then 106846168 [1, 2]. After border Vinogradov declined, but it has not yet reached number available for the calculation of modern computers. After reading the letter Goldbach, Euler noticed that the number “1” be not a prime number, and put forward a stronger assumption: every even number greater than two can be represented as the sum of two different or identical primes, i. e. N = 2t = pl + p2 . Further we will be called a couple of these primes (Pd P2) as a pair of Euler-Goldbach (PEG). Numerical testing in 2008 [3] confirmed the presence of pairs of Euler-Goldbach for all N < 1,2 ■ 1018. In the 25 volume encyclopedia of mathematical “World of Mathematics”
(2014) states that the hypothesis has not been proved analytically, and this problem is one of the most difficult problems of mathematics of the last two and a half centuries [3]. In the article Druzhinin [4] gave an analytic proof of fidelity the Euler-Goldbach conjecture. In this paper we present a different, more simple and profound proof of this hypothesis.
Suppose we have an even number N = 2t,t e N, t > 3. Since t can be odd or even number, we introduce the function a(t) = (l + (-l)j/2). If t is even, then a (t) = 1, and if t is odd, then a (t) = 0. Next, we define a basic set (BS) of a prime numbers for every even number N: ac (N) = { = 3;f2 =5; p3 = 7;...; pc }.Thelimiting prime pc determined from the ratio
(2+3)< N <( +1) (see. Table. № 1).
Table № 1. - Interval of even numbers (EN) and their basic sets
Interval EN [12,26] [28,50] [52,122] [124,170] [172,290]
BS a, ={3} II a 3 ={3;5;7} a 4 ={3;5;7;11} a ={3;5;7;11;13}
The meaning of the basic set is that all the odd constituent numbers d e [3,N - 3] are multiples of one or more primes from pt £ac (N). The fact is that the sieve of Eratosthenes d (ptk) = pt + f k, k <e N, k > 1, crossing out composite numbers, determines not only the first not crossed out number as prime pi, but at the same time creates the first buffer zone [pt, p2+1 - 2], in which all not deleted by set ai number is prime. For example, basic set a3 have a buffer zone [7,119].
Further in the analysis of N, we dispose all the odd numbers in the interval [3,N - 3] in a matrix of three rows and s = (t -1 -a(t)) / 2) columns. At the top row there are the odd numbers in the interval [3,2s +1] in ascending order. At the bottom (third) row there are odd numbers from the interval [2s +1 + 2a (t), N - 3] in reverse order. For example, when N = 56, we have a matrix (t = 28, number of columns s = 13):
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Section 4. Mathematics
' 3 5 7 9 11 13 15 17 19 21 23 25 27л
D(N)= 1 2 3 4 5 6 7 8 9 10 11 12 13
v53 51 49 47 45 43 41 39 37 35 33 31 29^
In this building matrix, sum of upper and lower
numbers in each column equal to N = 56. The second row numbers columns and is the indexes m xm = 2m +1 of the top row, where 1 < m < s. Thus, the matrix D (N) gives all possible combinations of an even number N for two odd numbers, except for the case N = 1 + (N -1)). We are interested in the columns, in which both the upper and lower numbers are prime. In this example such columns are three 56 = }1 + p2:};53),(13;43),(19;37)} . Since basic set a3 operates for N = 56, we consistently remove all the numbers at the top and bottom rows, multiple {3;5;7}, by the sieve of Eratosthenes. As a result, the matrix D (N) is transformed to
( 3 5 7 11 13 17 19 23
D (56) =
1 2 3 4 5 6 7 8 9 10 11 12 13 v53 47 43 41 37 31 29y
After this we will delete columns with two remaining numbers. There will be three ways to represent a number N = 56 as the sum of two primes. To simplify the process of comparing the top and the bottom rows, we present the process of removing the composite numbers at the second row. For this we introduce m (pi ) = (pi -1) / 2) for each pi from basic set and at the top row we make the arithmetic progressions
m (pt )= m (pt) + pt k, (1)
where 1 < k <( - m (pt))/ m (pt)) . There are three arithmetic progressions [(1 + 3k);(2 + 5k);(3 + 7k)f for D (56). At the second row, they removed the numbers {;7;10;12;13}, which exactly correspond to the component numbers at the top row.
To remove the composite number at the bottom (third) row to enter c arithmetic progressions for the second row. Arithmetic progressions have following form
m {p, )=P{p,) + Pi (2)
where the initial terms ofthe new arithmetic progressions p{p,) are found by solving the equation
(n - 2p(pt)-1):) ( :- multiplicity) and
(1 - p. )<p(pt) < 0. Thus, at the second row falls there are (2 ■ c) arithmetic progressions that cross out bad columns and leave only pairs of Euler-Goldbach.
For D (56), we obtain m(3) = -1> m(5) = °> m(7) = -4. We have six arithmetic progressions in this case, where k > 1:
I 1 + 3k I 2 + 5k I 3 + 7k 1-1 + 3k I 5k I-4 + 7k~l
At the second row D (56), these arithmetic progressions leave, only three numbers: 1; 6; 9, which accurately allocate pairs of Euler-Goldbach.
Pay attention to the feature of these (2 ■ c) arithmetic progressions. If N is a multiple of pi, then the arithmetic progressions for the given prime number coincide at the first and the third rows. Therefore at the second row, we have only one arithmetic progression. At that m(pt)-p(pt) = pt. This is a very important factor influencing the increase in the number of pairs of Euler-Goldbach. For example, when N = 24, there are c = 2,s = 5 {1;2;3;4;5} at the second row and we have four arithmetic progressions: }l + 3k},{ + 5k}, {-2 + 3k}, {-1 + 5k}.The first and third arithmetic progressions overlap. The numbers 2; 3; 5 remain,
i. e. there are three pairs of Euler-Goldbach: 24 = 5 +19 = 7 +17 = 11 +13. For bigger N = 26, there are c = 2, s = 6 {1; 2; 3; 4; 5;6} and also we have four different arithmetic progressions: {1 + 3k},} + 5k}, }k}, {-4 + 5k}. They leave: 2; 5, i. e. we have two pairs of Euler-Goldbach: 28 = 5 + 23 = 11 +17.
Our numerical calculations have fully confirmed the identity of the algorithm, where N < 1012.
Nevertheless, this fact does not explain the occurrence of pairs of Euler-Goldbach for any arbitrarily large even number. Consider the evidence. First we recall the Euler function
с f
ф(М = 3 ■ 5 ■ 7 •.... ■ p ) = МП 1
k=‘ I
Pk)
(3)
Here M is the product of c factorial consecutive primes. The function ф(М) gives number of numbers are not multiples any pt ean on the interval [1, M ], which we call the second buffer zone. In other words, if we remove all members of the arithmetic progression {p, ■ k}, k >1 on the interval [1, M ], then it will remain intact exactly ф(М) numbers. We have extended the notion ofEuler’s function, assuming that it is removed a few non-overlaping arithmetic progressions with the same difference, but with different initial numbers c {{+ p,} - 0}>{= ил... f on interval [1, M ]. There
are n = arithmetic progressions in all. In this case the
number Jof not removed numbers determined by generalized Euler function on the interval [1, M]
ф(и ) = m П
i - L
(4)
Example. There are three arithmetic progression: 1 + 3(k-1)}, 5(k-1)}, { + 5(k-1)} .Number of
not removed by these arithmetic progression numbers
equal
(=1
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Proof of euler-goldbach’s conjecture
*15>-K1 -111 - 2)-6
These are the numbers {3;5; 6; 8; 11;15}.
In our problem, there are two or one arithmetic progressions for each prime. The specificity of their task (1,2) gives such generalized Euler’s function
(5)
nM = npk 1 = m П
r f'
1 - f
-Hz; -1)-
-У у
Hence, on the interval [1, JV!] the probability of finding of not removed number equals
h (fk -1)
rn = (cp(M)/M ) = П
i - A
Jk У
M
(6)
At this stage, we can come back to the second row ofthe matrix D(N), for which we used (2 ■ c) arithmetic progressions and left not deleted columns, giving pairs of Euler-Goldbach. We can estimate the number ofsuch pairs by using the theory of probability. Indeed, since the entire second row of the matrix is in the range of the first buffer zone, then all not deleted basic sets ac are a primes and all ofthem in pairs define pairs ofEuler-Goldbach. Multiplying the length of the second row on the probability of finding no removed numbers, we will find the approximate number ofsuch numbers n (PEG; N), and hence we find the number ofpairs of Euler-Goldbach for the even number N.
n(PEG; N ) = s jnfl - f VS^M^ } (7)
[ k= I Pk J k=1 M J
Consider N = 56. The exact number of pairsof Euler - Goldbach equals n(PEG;N) = 3. Since f = f2 = 2, f = 1, then there is probability value n (PEG ;56 ) 2,47.
Further we neglect the second term in the curly brackets and represent the number of PEG for each N in the form
N - 2 P + ))p (.-fL) (8)
n (PEG; N ) = -
Jk У
In what follows, we neglect the second term in the curly brackets. Also compare the exact value of n (PEG, 24) = 3 with a probability value. Here a2 = {3;5}, N = 24 is a multiple of«3» and so n (PEG ;24 ) = 5 -(2 ■ 3/3 ■ 5) = 2 . This trend is n (PEG; N) > n(PEG; N) (the exact number of pairs of Euler-Goldbach greater than its probability value) is traced by us in all calculations and proved analytically.
Consider the probabilistic number of pairs of Euler-Goldbach for large N. We write a few obvious relationships.
ln (co) = Jln
k=1
(
( к
In 1 -A =-v
V Pi У k=1
2 7
= >
P. - 2 P,
1 Г f,V
VPi У
1 -2*.
Vk У
>-У t"
( 2Л
V , У
1
\Vi У
(9)
in (®)>-7X—•
k-1 pk
We use the asymptotic equality Legendre-Chebyshev [3]
c 1
E — = ln(Kc ln (pc )) . (10)
k=' pk
When limKc = K e Ж. Since ln(o) > -7ln(Kc ln(pc)), then at the second row ofthe matrix D(N) the probability of finding pairs of Euler-Goldbach equal
® >, , \ w • (11)
(c ln (Pc ))
The number of pairs of Euler-Goldbach on the theory of probability equals
c N
n(PEG; N) >-——— > . (12)
(c ln (pc )) 4 (( ln (pc ))
Since p2c < N, we can reinforce inequality
P 2
n(PEG; N) > Pc . (13)
4(cln (Pc ))
Right side of inequality increases monotonically and tends to infinity if pc tends to infinity. Since n(PEG;N) >n(PEG;N) > 1, then at least there is one pair of Euler-Goldbach for any large N. Thus, Euler-Goldbach’s conjecture is proved. At the same time we have proved the ternary Goldbach conjecture also. Subtracting an odd number of any prime, we get an even number consisting of two primes.
If N is a multiple of «3», in (8) the first multiplier is equal to (2/3) instead of (1/3) in the opposite case. This increases the number ofpairs of Euler-Goldbach for even numbers, which are multiples «3». This is confirmed experimentally.
Work was presented at the seminar at the Institute of Theoretical and Mathematical Physics, VNIIEF (Sarov, Russia).
References:
k=1
1. Dicson L. (E). History of the Theory of Numbers, v. II, CPC, NewYork, 1971.
2. Sizii S. V. Lectures on number theory, Moskow, 2007.
3. Joaquin Navarro. Elusive ideas and timeless theorems. World ofmathematics, vol. 25, p. 58, Moscow De Fgostini, 2014.
4. Druzhinin V. V. NTVP, 2014, №. 3, p. 14.
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