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Rule of Thumb Bounds in Goldbach’s Conjecture

Christopher Provatidis*, Emmanuel Markakis, Nikiforos Markakis

Department of Mechanical Engineering, National Technical University of Athens, Athens, Greece

Abstract: This paper determines proper factors for old and novel logarithmic functions previously used in asymptotic formulas, to make them conservative lower bounds for the “thumb-of-rule” estimation of the number of representations of an even number 2n as a sum of two odd primes (Goldbach’s conjecture). Numerical experiments up to 2n = 500,000 show that, in the graph of the number of prime-pairs versus 2n, the ratio of the ordinate of lowest “cloud” points over the aforementioned functions tends asymptotically to values between 0.61 and 0.74. One of the three formulas proposed takes the simple form 4n/[3(lnn)²], which is a conservative lower bound for the number of representations of an even number 2n as a sum of two odd primes.

Keywords: Goldbach’s conjecture, probabilistic number theory, lower bound

Received January 03, 2013; Revised February 23, 2013; Accepted February 25, 2013

Citation: Christopher Provatidis, Emmanuel Markakis, Nikiforos Markakis, Rule of Thumb Bounds in Goldbach’s Conjecture. American Journal of Mathematical Analysis, 1 (1). 8-13. January 2013. doi:10.12691/ajma-1-1-2

1. Introduction

Goldbach’s conjecture [1] states that “every even number greater than 4 can be written as a sum of two primes”, namely:

[picture] | (1)

where n>4 and p, q are prime numbers.

The set of primes is designated by. On this topic a detailed literature survey with over 300 papers has been recently reported [2]. Despite this large volume of work, the problem remains unsolved for 270 years, even after our study. In order to be consistent with [2], in this paper the even number 2n is denoted by N, that is .

As researchers could not find a rigorous proof, they turned their attention to develop the so-called “extended” Goldbach conjectures that deal with asymptotic formulas for the number of representations of every large even number as the sum of two odd primes. The first definite formulation appears to be due to Sylvester [3] in 1871, who proposed the function (2N/ln N) multiplied by a product of the ratio , where p is an odd prime divisor of N. Fifty years later, Hardy and Littlewood [4, pp. 32-33] showed that Sylvester’s formula leads finally to the function (N/(lnN)²) that was later proposed by them [4,p.32] in their monumental “Conjecture A”. As they claimed [4, p.33], there is no sufficient evidence to show how Sylvester was led to his result. In contrast, Hardy and Littlewood gave a strict proof concerning the representation of any large even number N as the sum of four odd primes (Theorem E) as well as of every large odd number N as the sum of three odd primes (Theorem D) [4, p.31]. The aforementioned Theorems E and D refer to the functions (N³/(lnN)⁴) and (N²/(lnN)³), that are in the form Nr⁻¹/(lnN)r, where the power r (4 and 3, respectively) is equal to the number of odd primes by the sum of which the number N is represented. Hardy and Littlewood claimed that their method fails when r=2 (i.e. in the representation as the sum of two odd primes) [4, p.32], but by analogy they conjectured that the function had to be (N/(lnN)²):

[picture] | (2a)

In addition, Hardy and Littlewood presented their “Conjecture B”, which refers to the number of prime pairs differing by k, and less than a large number n. The latter is a conjugate problem to “Conjecture A” [4, pp.42-44]. Closely related, even a very recent Internet lemma of a reliable website in mathematics [5], reproduces the “Conjecture B” option, that is when , the number of representations of an even number N = 2n as the sum of two primes is approximated by:

[picture] | (2b)

Equation (2b) is usually called the “extended” Goldbach conjecture and appears increased complexity since on top of the product of ratios it also requires the computation of an integral.

In both Eq(2a) and Eq(2b), or p runs through all odd prime divisors of N, and is the twin primes constant [6,7]:

[picture] | (3)

Concerning the product that appears in Eq(2a) and Eq(2b), this has been conceived on the basis of probabilistic considerations where Cramér has played an important role [8]. His theory, later improved by Granville [9], is capable of proving that when N (=2n) is a multiple of 6, the number of representations is, on average, twice as big as it is when N is not a multiple of 6. The importance of the latter is extensively discussed in [2]. In addition, we have to remark that the increased number of representations for multiples of 6 is a closely related fact with the remark made my Hardy and Littlewood [4, p.43] that, “there should be approximately equal numbers of prime-pairs differing by 2 and by 4, but about twice as many differing by 6”.

For the completeness of the literature survey, we have to mention that alternative theoretical predictions for much more general problems (e.g., the heuristic for the Hardy-Littlewood k-tuple conjecture [4,10] and the Bateman-Horn conjecture [11]) have been proposed. The interested author can also consult standard texts [7] as well as recent textbooks [12,13] and lecture notes [14] (among others), whereas very recent results have been reported by Helfgott [15].

Definition-1. Let us define the function as the number of ways (representations) that an even number (N = 2n, n = 3, 4, ...) can be written as the sum of two primes, p and q that fulfill Eq(1). In other words, it is the total actual number of pairs (p, q) that fulfill Eq(1).

In this paper we are intended to derive a “smooth” envelope of the discrete curve that represents the above defined function, so as no point of the graph “versus N” is found below this envelope.

Between many choices of candidate smooth and monotonically increasing functions, we make the observation that if the (time consuming) product is omitted in either of Eq(2a) or Eq(2b), we can immediately derive conservative lower bounds for the extended Goldbach conjecture. The problem now restricts to the determination of proper coefficients that should replace the twin primes constant (), if necessary.

Within this context, this paper is a numerical investigation to determine rule-of-thumb lower bounds, particularly for large even numbers. The computation restricts to even numbers less or equal to 500,000.

2. Methodology and Results

A MATLAB® code was developed to determine the actual number of representations of all even numbers as the sum of two odd primes.

Definition-2. We define the following three possible reference functions, which are candidate to approximate the abovementioned function G(N) of actual Goldbach representations:

[picture] | (4a)

[picture] | (4b)

[picture] | (4c)

It is noted that the function Li(x) is a build-in function in MATLAB.

The reason for choosing the above three functions is as follows. The function G₁ was established in [2] after probabilistic analysis in which the even “p is prime” had been considered independent on the even “q is prime”. As shown in [2], G₁ represents an average line (i.e. a particular fit curve) that closely follows the center of mass of the cloud of points that fulfill Goldbach’s conjecture on the G-N plane. Since the sequence of even numbers consists of consecutive cells in the form (6λ-2, 6λ, 6λ+2) where (as previously mentioned) for the middle number N=6λ the prime-pairs are, on average, twice as big as it is when N=6λ-2 or 6λ+2, while the centroid of the corresponding triangle (on the G-N plane) is very close to the graph of the G₁-curve, it is easily concluded that a factor 2/3 is to be used.

Moreover, the functions G₂ and G₃ were directly taken from Eq(2a) and Eq(2b), respectively. It is still unclear which of them has the best performance. A graph of these three functions in the domain under investigation is shown in Figure 1.

Figure 1. Comparison between the three functions under investigation

Then, the following three ratios were determined:

[picture] | (5)

The results for the first ratio () are presented in Figure 2, where it can be noticed that the minimum values of the ratio fluctuate near to the value 2/3 (see details in Table 1). By definition, the reference function corresponds to . Interestingly, although the aforementioned line is much closer to the bottom than top (see, [2]), there are 95035 cloud points beyond (38 percent) and 154963 ones below (62 percent) the graph of (not shown in Figure 2).

Figure 2. Convergence of ratio r₁ versus increasing even numbers 2n

Obviously, in order to ensure that all actual points from the cloud of Figure 2 are beyond the lower bound under investigation, for each of the three functions (G₁, G₂ and G₃) we have to choose the corresponding factor as small as possible. But since we are mainly interested in large numbers, we concentrate in the interval [3×10⁵, 5×10⁵] for which Table 1 dictates that proper values could be CF1 = 2/3, CF2 = 0.74 and CF3 = 0.61, where CF stands for the words “Correcting Factor”.

Table 1. Details for the minimum ratio “r₁” defined by Eq(4a)

Based on the above factors, the functions 2/3G₁, 0.74G₂ and 0.61G₃, which are candidate for being lower bounds, were found to fail in their role only for small even numbers, as shown in Table 2. Therein, the number of violations refers to those cloud points that are found to be below the candidate functions. It can be noticed that the function 0.74G₂ is probably the most suitable as appears the fewest violations.

Moreover, the quality of the three proposed “lower bound” functions, to be used as lower bounds, is clearly shown in Figure 3, Figure 4, and Figure 5, respectively.

Table 2. Number of violations for the three lower bound curves

Figure 3. The function 2/3G₂ as a lower bound

Figure 4. The function 0.74G₂ as a lower bound

Figure 5. The function 0.61G₂ as a lower bound

As has been previously mentioned [2,16,17,18,19], the main reason for the factor 2/3 is the fact that even numbers in the form 2n=6k are represented by approximately a double number of prime-pairs (p,q) than what happens for their neighboring even numbers 6k-2 or 6k+2. For example, 6k = 90 is represented by 9 prime-pairs, while each of 6k-2 = 88 and 6k+2 = 92 are represented by only 4 prime-pairs.

Definition-3. Let us now call Tk (k = 1, 2, …) the arbitrary triangle determined by the aforementioned even numbers in the graph “number of prime-numbers versus the even number N = 2n”.

A careful examination until 2n = 500,000 has shown that the centroids of the successive triangles Tk closely follow the function . For example, Figure 6 shows that in the area of the even number 2n = 285,368 (where the last violation for the function 2/3G₁ occurs; see also the second column in Table 2), the centroids (connected by straight segments in magenta color) are very close to the function (in green color). It can be also noticed that the magenta line intersects the green line from place to place. In the same graph one can see details concerning the lower bound (2/3G₁), again shown in red color as was the case in Figure 3. In this interval, the only intersection of the blue line with the red one is the value 2n = 285,368. From the latter value until 500,000 no violation was found.

Figure 6. Number of prime-pairs (p,q) that fulfill Goldbach’s conjecture (in blue), centroids of triangles Tk (in magenta), function G₁ (in green) and lower bound 2/3G₁ (in red color); in the neighborhood of the even number 2n=285,368 where the last violation occurs

3. Conclusion

The numerical results of this study suggest that all three expressions found may be useful for a rule-of-thumb fast estimation of the number of representations of large even numbers as a sum of two odd primes. Of course, further research should be conducted for still larger even numbers but no serious changes are anticipated.

References

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[3]   | Sylvester, J. J., “On the partition of an even number into two primes,” Proc. London Math. Soc., s1-4(1). 4-6. 1871.
[4]   | G.H. Hardy, J.E. Littlewood, “Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes,” Acta Math., 44 (1). 1-70. 1923.
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[14]   | Koukoulopoulos, D., MAT 6684W: Sieve methods, Université de Montréal, Fall 2012.
[15]   | Helfgott, H.A., “Minor arcs for Goldbach’s problem,” May 2012.
[16]   | Markakis, E., Provatidis, C. and Markakis, N., “Some issues on Goldbach Conjecture,” 29 May 2012.
[17]   | Provatidis, C.G., “More properties in Goldbach’s Conjecture,” 22 October 2012.
[18]   | Provatidis, C.G., “Smoothening Properties Related to the Goldbach’s Conjecture,” 13 October 2012.
[19]   | Provatidis, C.G., “On the minimum number of pairs that fulfill Goldbach’s Conjecture,” 13 October 2012.