Generalization
of the Goldbach-Conjecture and Estimation of the Number of Solutions
Fritz Gassmann (PSI)
and
A. Marques Filho
(Instituto Nacional de Pesquisas
da Amazônia, P.O. Box 478, MCT/INPA, Manaus, AM-Brazil)
Abstract: We develop an asymptotically
exact analytical formula for the prime density function based on the
sieve process of Erathostenes. In contrast to the prime number theorem
introduced by Gauss, our formula can be generalized to include a finite
set of conditions C to be imposed on sums or differences of primes. With
one additive condition (C=1), the formula gives an approximation for the
number of Goldbach partitions and at the same time a relation for the average
lower bound of this number. The relative errors of the formula decrease with
increasing even number E and so enable it to explain the structure of Goldbach
partitions as e.g. the familiar "Goldbach-Comet", special E with large or
small numbers of partitions, etc. With one subtractive condition, the number
of prime pairs with a given distance (e.g. twin primes with distance 2) in
the interval 2...Z (Z=natural number) can be estimated. With C>1, many
additive and subtractive conditions can be mixed (e.g. for investigation of
equidistant primes). The formula defines also the restrictions imposed on
sums or differences of primes to ensure abundant solutions. Our method is
based on an equality assumption we cannot prove (a proof would be equivalent
to a proof for the Goldbach conjecture). However, reasons given for its validity
include an observed scaling relation for the quasi-random errors of the formula,
suggesting a binomial error distribution with a limited number of steps leading
to a positive minimum number of Goldbach pairs for every E.
Keywords: Goldbach Conjecture, Generalized Goldbach Problems, Estimation
of Number of Goldbach Partitions
Goldbach's
conjecture states that every even integer E>2 can be expressed
as sum of two primes (in the following called Goldbach pairs or Goldbach
partitions). Though computer-based numerical tests have confirmed the
theorem up to 4·1014, it resisted any formal mathematical proof
up to now . A necessary condition for the conjecture to be valid is, that
new primes steadily appear on the sequence of integers without limit.
The respective proof was already given by Euclid around 300 BC and has
been confirmed repeatedly in different forms since then.
We performed previous studies on the structure of prime-multiples and
its effects on Goldbach pair frequencies (Marques Filho & Walker,
2001) and a series of numerical tests up to E = 3·107 confirmed
the developed analytical approximation for the number of Goldbach pairs
(Marques
Filho, Gassmann & Walker, 2005). They showed the relative error
of the approximation decreasing proportional to about E-0.4
for increasing E. Further numerical tests up to E = 2.56·1010
showed converging of this power law towards E-1/2.
In the present paper, we will generalize our approach. We begin with a
slightly modified sieve process to develop an asymptotically exact relation
for the prime density. A generalization of this relation on the basis
of an equality assumption (to be defined later) will unite the sets of
primes, Goldbach pairs, twin primes and higher prime combinations to members
of a family called generalized Goldbach problems. To demonstrate our procedure,
we then apply it to the special case of Goldbach pairs and develop a
calibrated formula for the number of Goldbach partitions for every even
number E up to at least 2.56·1010. After showing other applications,
we investigate deviations of our relation from the true number of Goldbach
pairs and show them to scale with the square root of the number of partitions.
On this basis, we present reasons for the validity of our equality assumption.
Its proof would be equivalent to a proof of the Goldbach conjecture and
so is beyond the scope of the present work. However, a heuristic demonstration
of the truth of the Goldbach conjecture, based on the observed scaling
law, is given at the end of this paper.
Formula for the estimation of the number
of Goldbach partitions:
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Numerical tests for even
numbers E performed by F. Gassmann up to E=2.56·1010 reveal this
calibrated approximation for the number of Goldbach partitions with
alpha ≈ 0.5 egamma ≈ 0.890536
gamma = Euler constant ≈ 0.577215
a ≈ -0.5455
b ≈ 0.5153
The product runs over all primes P'. The function delta(P') has only
the two possible values 1 (E is a product of P') and 2 (E is not a product
of P'). The asymptotic prefactor alpha2 was found to be the
best approximation for large E. a and b were calibrated to give good results
for the whole range from 4 up to 2.56·1010. Deviations of this
approximation from the exact number of Goldbach pairs never exceeded 3.61·NG1/2
for all E up to 1'000'000. This observation suggests scaling of absolute
errors with NG1/2, leading to vanishing relative errors
proportional to NG-1/2, a signiture of Poisson statistics
often found in physics for rare events (e.g. radioactive decay).
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Fig. 1: "Goldbach Comet" near E=3·107.
Our approximation NG (squares) is compared with the exact numbers
of Goldbach partitions (points). The horizontal line at 151'869 indicates
the lower bound according to our estimate. The squares encompass about
±10·NG1/2. Maximum errors in this range are ±2.4·NG1/2.
The branch near 300'000 stemming from P'=3 contains E's being multiples
of 3 and lies a factor of 2 over the well-defined lower boundary (for detailed
explanations see full
text).
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Fig. 2: Natural logarithm of the normalized error distribution
for 110'513 even numbers E between 500'000 and 721'024. Errors are defined
as deviations (NG-NG, exact) of the number of Goldbach
partitions NG calculated with the above formula from the exact
number NG, exact based on computerized counting. The abscissa
indicates normalized errors. The squares are calculated numbers of errors
within intervals of width 0.1. The blue line indicates respective values based
on a Gaussian distribution with the same normalized standard deviation 0.736.
The mean lies at -0.216 and was shifted to 0 for comparison with the Gauss
distribution. Higher moments are 0.0564 (skewness) and 0.0188 (excess)
which both vanish for a Gauss distribution. The maximum deviation occurred
for E=712'432=24·7·6361 and was 4.6 normalized standard deviations (a second
error was near to this value, both together leading to the square at the
bottom left). These two points are the largest errors existing below 106.
The calculated distribution is not distinguishable from a Gaussian distribution
within the range of about 5 standard deviations.
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