Goldbach Conjecture

Maths

What different approaches have been applied in attempts to mathematically prove and verify that Goldbach’s binary and ternary

conjectures are true?

Name: Anil PrasharWord count: 3718 wordsAbstract Word Count: 211 words

IB Extended Essay: Maths Anil Prashar

Abstract

This essay attempts to compare and analyse some of the successful, classical methods,

as well as some of the new more interesting approaches that have been applied in the

attempt to prove Goldbach’s conjecture over time. There have been various different

approaches from mathematicians of varied calibres, each either furthering the

conjecture or helping to change the way the conjecture is viewed.

I attempt to consider and compare different approaches not strictly with the use of

rigorous proof as most ideas are presented in maths. Instead I compare the key points

of each idea and method, whilst considering it in terms of the initial conjecture. The

first real attempts to analyse Goldbach’s conjecture were in the 1900’s. In this essay I

attempt to track the progress of number theory, by looking at methods that have been

applied in the last century.

The question is of significance as it considers two fairly fundamental ideas in

mathematics, in the sense that; many amateurs and mathematicians find the conjecture

easy to understand. I will therefore be approaching the conjecture from the position of

an amateur, with the knowledge of a college student. Understandably, this makes

some ideas inaccessible to me, but this essay attempts to understand the key points

behind theorems and ideas involved.


Table of Contents

I) Introduction......................................................................page 3

II) The Binary and Ternary Conjectures; Linked..............page 4

III) Proof by Probability.........................................................page 6

IV) Schnirelmann’s Theorem…………………………..….page 8

V) The Smallest Partition………………………………....page 11

VI) Sieve Theory………………………………………...….page 14

VII) Graphical methods and Vinogradov’s Theorem…….page 18

VIII) Conclusion………………………………………...……page 21

IX) Bibliography………………………………………..…..page 22


I) Introduction

In 1742 the amateur mathematician, Christian Goldbach, wrote a letter to the

infamous Leonhard Euler. The letter suggested a possibility regarding number theory;

it would later come to be known as ‘Goldbach’s Conjecture’. This conjecture stated

two major ideas:

1. Every even integer 6 can be expressed as the sum of two positive, prime

numbers; often named Goldbach’s binary (strong) conjecture.

2. Every odd integer 9 can be expressed as the sum of three positive, prime

numbers; often named Goldbach’s ternary (weak) conjecture.1

Part 1 means that 6, 8, 10, 12 etc. (all even integers) can be displayed as the sum of

two primes. Examples as to why this may be true are shown below:

Part 2 states that 9, 11, 13, 15 etc. (all odd integers) can be displayed as the sum of

three primes. Examples are shown below:

Euler agreed with Goldbach’s findings, but stated that he was unable to find the proof

himself. The binary and ternary conjectures remain unproved to this day. The

conjecture is, perhaps, particularly interesting as there have been numerous

approaches to the Goldbach conjecture based on the state of number theory. By

exploring how different methods have been applied over time, one can see the

development of number theory in its attempts to prove conjectures.

1 Wang, Yuan. The Goldbach Conjecture. London: World Scientific Publishing Company; 2 Sub Edition, 2003.


II) The Binary and Ternary Conjectures; Linked

Let N = a sufficiently large positive, even integer

p1 and p2 = two prime numbers (may be identical)

If

(i.e. If Goldbach’s binary conjecture is true)

Then

2

This therefore suggests that if Goldbach’s binary conjecture is correct then the ternary

conjecture is proved by default. This is because an even number plus three is odd. In

addition to this an even number plus any other prime number (excluding 2) is always

odd, and therefore p3 could just as easily have represented 5, 7, 11, 13 etc; thus

creating further combinations for the composition of an odd number.

I also consider this idea in a different light. For example if the ternary conjecture were

able to be proved on its own merit, then even numbers could be shown to be the sum

of 4 primes as opposed to 2 (shown below).

Let O = a sufficiently large positive, odd integer

p1, p2 and p3 = three non-identical prime numbers

If

(i.e. if Goldbach’s ternary conjecture is true)

Then

2 Shi, Kaida. "A New Method to Prove Goldbach Conjecture, Twin Primes Conjectures and Other Two Propositions." Zhejiang Ocean University, 2000: 2.


As any odd number plus another odd number is always even, then all even numbers

could be shown to be the sum of 4 primes. However, both of these ideas are

dependent on the massive assumption that either the binary or ternary conjectures are

already true. As the binary automatically proves the ternary it is only natural that it

has been subject to greater attention i.e. it has had more attempted proofs. It is now

time to consider the conjectures themselves. The reason as to why the integers had to

be ‘sufficiently large’ is explained later in part VI [Sieve Theory].


III) Proof by Probability

Some mathematicians conjecture that if the binary conjecture is relatively simple and

easy to understand, then perhaps the proof shares a similar simplicity. This has led to

some people accepting Goldbach’s theory under the heuristic method of probability.

Probability is generally an unaccepted method of proving a conjecture in maths;

however it seems to be one of the methods used to enhance the idea that Goldbach’s

conjecture is correct. The process of thought is as follows:

As the even integer that is being considered increases, the number of partitions also

increases:

This is perhaps further clarification that as we continue to verify larger even numbers

are the sum of two primes our chances of finding one that isn’t the sum of two primes

diminishes. Thus far the Goldbach conjecture has been verified by computers to be

correct until suggesting that now a pattern or proof is to be discerned3 .

A particularly distinct approach has been used by the mathematician, Mark

Herkommer; he begins by examining the probability of prime pairs occurring in the

partition of 100 based on intervals of size ten4:

n = 100

Interval % prime Matching Interval % primeprobabilityprime pair

Probabilityneither prime pair

0 - 10 60 90 – 100 20 0.12 0.8810 - 20 80 80 – 90 40 0.32 0.6820 - 30 40 70 – 80 60 0.24 0.7630 - 40 40 60 – 70 40 0.16 0.8440 - 50 60 50 – 60 40 0.24 0.76

% prime: Shows the percentage of the odd numbers in the interval that are prime.

3 T., Oliveira e Silva. Goldbach Conjecture Verification. July 14, 2008. http://www.ieeta.pt/~tos/goldbach.html (accessed July 1, 2009).4 Herkommer, Mark. Goldbach Conjecture Research. May 24, 2004. http://www.petrospec-technologies.com/Herkommer/goldbach.htm (accessed June 30, 2009).

http://www.ieeta.pt/~tos/goldbach.html


The probabilities of a prime pair summing to make 100 are calculated by multiplying

the percentage of prime numbers in the ‘Interval’ and the percentage of prime

numbers in the ‘Matching Interval’.

To calculate the probability of a prime number partition being possible for 100, you

multiply all the probabilities of the event not happening. Then as there are only two

possibilities; either the event happening or not happening. The probabilities must sum

to make 1 and ‘1 – the total probability of the event not happening’, in this case, gives

you the probability 0.70967.

If this process is then repeated on larger values, then the probability can be found to

increase as shown below5:

n probability

1000 0.996208045988

2000 0.999838315754

3000 0.999999069064

4000 0.999999693974

5000 0.999999983603

6000 0.999999999995

7000 0.999999999875

8000 0.999999999978

9000 1.000000000000

10000 1.000000000000

However, as the author of the method recognises, the larger probabilities occur when

smaller intervals are used. He also appreciates that there may be the one number that

is not noticeable through such an imprecise method. The main point of this idea is to

recognise that Goldbach’s conjecture, in terms of probability, is most likely true.

IV) Schnirelmann’s Theorem

5 ibid.


‘There exists a positive integer “s” such that every sufficiently large integer is the

sum of at most “s” primes. It follows that there exists a positive integer such that

every integer is a sum of at most primes. The smallest proven value of is

known as the Schnirelmann constant.’6

The magnitude of this theorem meant that Goldbach’s conjecture was finally to be

brought into the finite realms of mathematical investigation. The proof is based on

Mann’s theorem. Mann’s theorem states that if ‘A and B are sets of integers each

containing 0’, then:

‘ ’7

represents the Schnirelmann density, defined as the ‘greatest lower bound of the

fractions where is the number of terms in the set ’8.

where Q represents a set.

e.g. Set: {0, 2, 4, 6, 8}

When ,

In this situation, trial and error was used to calculate the Schnirelmann density; the

greatest lower bound is the smallest value of which in this case was < .

Greatest Lower Bound: If it is given the value in set , then . This

could otherwise be seen as .

represents the direct sum, where each element of set

A is added to every element of set B, any repeated values are discarded.

represents the smallest value in the set A or B, depending on which

one is smaller.

So to simplify the previous points and bring it back into the context of Schnirelmann’s

Theorem, Mann’s theorem states that the greatest lower bound of the sum of two sets

is greater than or equal to the greatest lower bound of set A plus the greatest lower 6 O'Bryant, Kevin. Schnirelmann's Theorem. http://mathworld.wolfram.com/SchnirelmannsTheorem.html (accessed July 2, 2009).7 O'Bryant, Kevin. Mann's Theorem. http://mathworld.wolfram.com/MannsTheorem.html (accessed July 2, 2009).8 Weisstein, Eric W. Schnirelmann Density. http://mathworld.wolfram.com/SchnirelmannDensity.html (accessed July 2, 2009).

http://mathworld.wolfram.com/SchnirelmannConstant.html

http://mathworld.wolfram.com/PrimeNumber.html

http://mathworld.wolfram.com/Integer.html

http://mathworld.wolfram.com/PositiveInteger.html

http://mathworld.wolfram.com/PrimeNumber.html

http://mathworld.wolfram.com/Integer.html

http://mathworld.wolfram.com/SufficientlyLarge.html

http://mathworld.wolfram.com/PositiveInteger.html


bound of set B. In terms of Schnirelmann’s theorem, one very important point to

remember is that , as mentioned earlier.

Let

Where p is all prime numbers

By using the direct sum method and discarding all repeated values, you get:

It can be shown using the inclusion-exclusion principle9 that:

But

Now if we consider that is the greatest lower bound of , and is , then the

following method applies:

Remember that as , the maximum value that

can ever take, depending on how many times P is added, will be 1. Therefore if this

process is then repeated using , then , this leads to:

(This is considered an acceptable rule)

10

This is because eventually , therefore the density = 1. This is clearly

noticeable in set Q, as it has many more numbers, causing its Schnirelmann density to

increase. It is also only possible for sets to have Schnirelmann density = 1, if and only

9 Inclusion-Exclusion Principle. June 26, 2009. http://en.wikipedia.org/wiki/Inclusion-exclusion_principle (accessed July 1, 2009).10 Schnirelmann's Theorem. June 27, 2009. http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem (accessed July 1, 2009).


if the set contains all positive integers, because then at every value.

Therefore if the process is repeated times on the initial prime number set, then k can

be used to deduce the maximum amount of prime numbers that sum to make any

positive integer. This process is intensely intricate and is carried out through the use

of various set theory principles. With it Schnirelmann was able to find the number s0,

which represented the minimum amount of prime numbers required to represent the

sum of each number. Schnirelmann initially found the constant was 300000 prime

numbers11. This was improved over time, as shown below12:

s0 Author

159 Deshouillers (1973)

115 Klimov et al. (1972)

55 Klimov (1975)

27 Vaughan (1977)

26 Deshouillers (1977)

19 Riesel and Vaughan (1983)

7 Ramaré (1995)

As can be seen from this table, Ramaré has developed the idea furthest with his

findings, using Schnirelmann’s initial theorem and sieve theory to prove that at most 7

prime numbers are required to represent all numbers. Although this is close to

Goldbach’s initial conjectures, it is not quite there yet.

V) The Smallest Partition

When considering the Goldbach conjecture, one large difficulty is the fact that there

are numerous partitions for each number making patterns more difficult to discern. In

11 Hofstadter, Douglas R. Goedel, Escher, Bach; an Eternal Golden Braid. London: Penguin Books Ltd., 2000.12 Weisstein, Eric W. Schnirelmann Constant. http://mathworld.wolfram.com/SchnirelmannConstant.html (accessed July 2, 2009).


this situation some mathematicians have approached the numbers looking at where the

smallest prime number partition exists. The results for n<1000000000 are as below13:

n g(n) n - g(n) g(n) / n6 3 3 0.500000000

12 5 7 0.41666666730 7 23 0.23333333398 19 79 0.193877551

220 23 197 0.104545455308 31 277 0.100649351556 47 509 0.084532374992 73 919 0.073588710

2642 103 2539 0.0389856175372 139 5233 0.0258749077426 173 7253 0.023296526

43532 211 43321 0.00484700954244 233 54011 0.00429540663274 293 62981 0.004630654

113672 313 113359 0.002753536128168 331 127837 0.002582548194428 359 194069 0.001846442194470 383 194087 0.001969455413572 389 413183 0.000940586503222 523 502699 0.001039303

1077422 601 1076821 0.0005578133526958 727 3526231 0.0002061273807404 751 3806653 0.000197247

10759922 829 10759093 0.00007704524106882 929 24105953 0.00003853727789878 997 27788881 0.00003587637998938 1039 37997899 0.000027343

113632822 1163 113631659 0.000010235187852862 1321 187851541 0.000007032335070838 1427 335069411 0.000004259419911924 1583 419910341 0.000003770721013438 1789 721011649 0.000002481

n = a random even integer between 0 and 1000000000

g(n) = the smallest prime number that sums with another prime number to make n

= Due to simple addition this becomes the largest prime number that sums to make n with another prime.

= An interesting column that shows the ratio between n and the smallest partition value.

13 (Herkommer 2004)


One evident observation in the column is that the value of this is ratio is

decreasing. This idea is graphically portrayed below14:

Where X-axis = log10n [used to help show numbers on a closer scale]

Y-axis = g(n)

As is evident from the graph, there is a clear exponential incline that can be discerned

from the graph. Below I have added a curve of best fit to the graph to analyse it more

efficiently:

14 ibid.


Perhaps the most striking detail about the curve is that most of the points do not lie on

it. This therefore means that there isn’t a consistent increase in the ratio of the

smallest partition to the initial number; in essence nullifying the method. It appears

that all partitions need to be considered if the conjecture is to have a consistent proof.

VI) Sieve Theory

You may be aware of the sieve of Eratosthenes, a method used to find prime numbers

by cancelling out multiples of a number as you continue to move across a number


line. In 1915, the mathematician Viggo Brun developed a new type of sieve, known

today as Brun’s sieve.

Brun’s sieve is an estimation method15, where the sizes of ‘sifted sets’ are estimated

based on congruences between groups based on set conditions. A congruence ‘on a

set X determines a partition of the set X to which it corresponds’16. In simpler terms

this means that when a certain condition is applied to a set, a congruence is what

determines each of the subsets created by the condition.

A simple example might be where a large set (all numbers in this case) is taken, and

all even numbers are taken and placed in a subset; the congruence is the determination

that one subset is all even numbers. The odd numbers are also placed in another

subset. Now if I took specific elements from the original set (all numbers), in this case

I shall take 7 and 16, they are only considered equivalent based on the subset that they

are placed in. As 7 is odd and 16 is even, they are not considered equivalent. However

if I had taken 10 and 16 they would have been considered equivalent based on the fact

that they are in the same subset.

To discuss this concept further, it is first necessary to go back to Euler and his works.

One of the key observations that Euler had made in his works was that the sum of the

reciprocals of all prime numbers diverges17. It was Brun who made an even more

interesting observation however. By using the observation that:

18

is defined as , where p and q are both prime. The

mathematical term for these types of numbers is ‘twin primes’.

The ‘O’ function is the Landau notation. This can be described as follows:

As

15 Brun Sieve. May 19, 2009. http://en.wikipedia.org/wiki/Brun_sieve (accessed June 27, 2009).16 Eccles, Peter J. An Introduction to Mathematical Reasoning; numbers, sets and functions. Cambridge: Cambridge University Press, 2001.17 Caldwell, Chris K. There are infinity many primes, but how big of an infinity? http://primes.utm.edu/infinity.shtml (accessed June 30, 2009).18 Charles, Denis Xavier. "Sieve Methods." 2000: 35.


But this is only if for a large value of there exists positive real integers for M and

such that:

For all values of

Where M is a constant19

This basically means that the Landau notation is used to help express as

tends to infinity in the form of a simplified .

For example:

Let

Now we want to consider the behaviour of as . It is obvious that will

grow the fastest, as it has the highest derivative ( ).

[Remember ]

Now this needs to be in the same form as , where M and are

positive, real integers.

Let

[Modulus has been taken]

[ ]

[Collect like terms]

Thus to use the original form:

For all

In this particular circumstance, as the twin primes tend to infinity, they have been able

to be expressed in a simpler form using Landau notation. It was with this form that

Brun proved that the sum of all twin primes converges. As he found that the value for

the sum of the reciprocals of all twin primes was around 1.90216054, a number now

known as ‘Brun’s Constant’. Its finding meant that there were believed to be a finite

amount of twin primes.

19 Big O Notation. June 28, 2009. http://en.wikipedia.org/wiki/Big_O_notation (accessed July 1, 2009).


In consideration of Goldbach’s conjecture, we are working with the assumption that

there is an infinite amount of numbers. Therefore for Goldbach’s binary and ternary

conjectures to be true there must be an infinite amount of prime numbers. It was

proved true by Euler that there were an infinite amount of prime numbers20. However

the suggestion that there are a finite amount of twin primes has possible implications

for Goldbach’s conjecture, as certain numbers are only expressible as the sum of twin

primes e.g. etc.

On the other hand, going back to part III [Proof by Probability], larger numbers

generally have more partitions, and therefore twin primes may be considered to have

no relevance to Goldbach’s conjecture (except at early stages) e.g. 24 is the sum of the

twin primes 11 and 13, but is also the sum of 5 and 19.

Hardy also suggested a further link between twin primes and Goldbach’s conjecture

by suggesting that the function G(N) [which represents the number of ways in which

N can be written as the sum of two primes] was asymptotic to some function of the

twin prime constant21. The twin primes constant is defined as:

Where p is a prime number

A solid link between Goldbach’s conjecture and twin primes was validated by Chen

Jingrun in 1966, using an extremely long and rigorous proof; with the help of sieve

theory. Some very interesting and important points of consideration came from his

proof22, where he deduced that all sufficiently large even numbers (represented as N)

could be partitioned in two possible ways:

1)

20 Caldwell, Chris K. Euclid's Proof of the Infinitude of Primes. http://primes.utm.edu/notes/proofs/infinite/euclids.html (accessed June 29, 2009).21 Caldwell, Chris K. Goldbach's Conjecture. http://primes.utm.edu/glossary/xpage/GoldbachConjecture.html (accessed June 29, 2009).22 PrimeFan. Chen's Theorem. http://planetmath.org/encyclopedia/ChensTheorem.html (accessed June 27, 2009).


2)

(Where p1, p2 and p3 are all prime numbers)

Form 1) was mentioned in part II of this essay [The Binary and Ternary Conjectures;

Linked]. This is an important observation, for Chen has perhaps come closest to

proving the binary conjecture with his sieve method. It has already been discussed

how Chen’s form 1) would prove the ternary conjecture by default.

However form 2) recognises that some even numbers have been sorted as the sum of a

prime and a semi-prime. A semi-prime is defined as the product of two prime

numbers. Thus the conjecture still remains unproved. It is perhaps possible, using

more refined methods of ‘sieving’ and estimation to finally prove Goldbach’s

conjecture, yet such methods have not been attempted as of yet.

VII) Graphical methods and Vinogradov’s Theorem

An interesting method pursued by some mathematicians is the use of graphs23:

23 (Herkommer 2004)


The graph shown is known as ‘Goldbach’s Comet’, a few propositions that have been

made in this essay become explicitly clear through the use of this graph. For example

the number of partitions increases as the number being considered increases (Part III

[Proof by Probability]).

The main point of using a graph is based on its curving shape. It has been suggested

that the proof may be dependent on some sort of asymptotic approach in order to help

prove the conjecture. This approach was advocated by various renowned

mathematicians in their attempt to tackle Goldbach’s conjecture e.g. Ramanujan,

Hardy, Erdös etc.

However, it was to be the Russian mathematician, Ivan Vinogradov, who would

effectively find the method that has made one of the most important leaps in helping

to prove Goldbach’s ternary conjecture. He found that every ‘sufficiently large’ odd

integer could be expressed as the sum of three primes. He did indeed use asymptotic


analysis to give finite bounds for the number of partitions that could be made of odd

integers as the sum of three primes. His theorem is as follows:

Let A = a positive, real integer

N = a sufficiently large odd integer

p1, p2 and p3 = three primes that sum to make ‘N’

Then according to Vinogradov’s theorem:

24

represents the function of N that is used to represent Vinogradov’s theorem.

G(N) represents the number of ways in which N can be partitioned as the sum of a

certain amount of primes.

You may once more recognise the (O) Landau notation, used to show r(N) as

. This is a quantity that is required in order to assess r(N) asymptotically (as

it tends to infinity).

And:

Where:

(n) {known as the Von Mangoldt function} =

[If where p is a prime and a is an integer ≥ 1]0 [If ] 25

Therefore:

24 Vinogradov's Theorem. June 24, 2009. http://en.wikipedia.org/wiki/Vinogradov%27s_theorem (accessed June 27, 2009).25 Von Mangoldt function. May 2, 2009. http://en.wikipedia.org/wiki/Von_Mangoldt_function (accessed June 30, 2009).


Using certain rules and techniques it can be shown that:

Directly following on from this statement, it was found that when N was odd, then

G(N) was approximately 1 (the number of ways of partitioning a number as a sum of

primes was 1). After further rigorous analysis, the key point of attention is that:

The number of ways N can be written as the sum of 3 primes

This consequence of the equation suggests that when asymptotic analysis is used, a

proof to Goldbach’s ternary conjecture can be achieved. The only problem is that the

theory is only applicable to ‘sufficiently large’ numbers. It was later specified by one

of Vinogradov’s students that greater than 314348907 was sufficiently large enough26.

However, this is ridiculously large, as computerised methods have only checked

numbers up to 18. Meaning that there is an extremely large range of numbers

that have not been checked to see if they are the sum of three primes or not. Thus the

ternary conjecture remains unproved.

VIII) Conclusion

26 Goldbach's Weak Conjecture. June 24, 2009. http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture (accessed June 28, 2009).


There are various ways to express a problem in mathematics. Through proper

manipulation and development, a specific formula will allow for a proof. Some forms

of the Goldbach conjecture have been lesser developed. There have been numerous

methods of the sort. The methods explored in this essay have been either some of the

simpler or more successful methods applied.

The simple methods, such as probability, graphical methods and the smallest partition

were useful ways of helping to look at ways in which conjectures may appear to be

true. By examining that the probability of the Goldbach conjecture being true

increased for larger values, the idea was enhanced that the theory was almost

indefinitely true. However, there existed obvious flaws in the method. Probability can

generally be summarised as a situation where something does or doesn’t happen (both

can’t happen together). As the probability is never 1 for any of the partitions (although

the larger values do tend to it) there is still a chance that there is one renegade value

that would disprove Goldbach’s conjecture. Graphical methods proved useful in

spotting links between certain parts of Goldbach’s conjecture, but the smallest prime

number partition idea was of no use.

The successful methods (i.e. the more rigorous methods) such as Schnirelmann’s

theorem, Brun’s sieve, and Vinogradov’s theorem yielded much more beneficial

results. Schnirelmann’s theorem proved that the conjecture was true in the respect that

all numbers could indeed be split into the sum of a certain amount of prime numbers.

In addition to this, Brun’s sieve and Vinogradov’s theorem helped to partially verify

Goldbach’s ternary and binary conjectures, although the theorems were still only

applicable to sufficiently large numbers. The magnitude of sufficiently large meant

that the theory would have to be proved by computerised methods up to these values.

A concept that mathematicians should be cautious of, as verifying a conjecture is one

thing, but understanding why it is true is dependent on rigorous proof and analysis.

IX) Bibliography


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Brun Sieve. May 19, 2009. http://en.wikipedia.org/wiki/Brun_sieve (accessed June 27, 2009).

Caldwell, Chris K. Euclid's Proof of the Infinitude of Primes. http://primes.utm.edu/notes/proofs/infinite/euclids.html (accessed June 29, 2009).

—. Goldbach's Conjecture. http://primes.utm.edu/glossary/xpage/GoldbachConjecture.html (accessed June 29, 2009).

—. There are infinity many primes, but how big of an infinity? http://primes.utm.edu/infinity.shtml (accessed June 30, 2009).

Charles, Denis Xavier. "Sieve Methods." 2000: 35.

Eccles, Peter J. An Introduction to Mathematical Reasoning; numbers, sets and functions. Cambridge: Cambridge University Press, 2001.

Goldbach's Weak Conjecture. June 24, 2009. http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture (accessed June 28, 2009).

Herkommer, Mark. Goldbach Conjecture Research. May 24, 2004. http://www.petrospec-technologies.com/Herkommer/goldbach.htm (accessed June 30, 2009).

Hofstadter, Douglas R. Goedel, Escher, Bach; an Eternal Golden Braid. London: Penguin Books Ltd., 2000.

Inclusion-Exclusion Principle. June 26, 2009. http://en.wikipedia.org/wiki/Inclusion-exclusion_principle (accessed July 1, 2009).

O'Bryant, Kevin. Mann's Theorem. http://mathworld.wolfram.com/MannsTheorem.html (accessed July 2, 2009).

—. Schnirelmann's Theorem. http://mathworld.wolfram.com/SchnirelmannsTheorem.html (accessed July 2, 2009).

PrimeFan. Chen's Theorem. http://planetmath.org/encyclopedia/ChensTheorem.html (accessed June 27, 2009).

Schnirelmann's Theorem. June 27, 2009. http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem (accessed July 1, 2009).

Shi, Kaida. "A New Method to Prove Goldbach Conjecture, Twin Primes Conjectures and Other Two Propositions." Zhejiang Ocean University, 2000: 2.

T., Oliveira e Silva. Goldbach Conjecture Verification. July 14, 2008. http://www.ieeta.pt/~tos/goldbach.html (accessed July 1, 2009).

http://www.ieeta.pt/~tos/goldbach.html


Vinogradov's Theorem. June 24, 2009. http://en.wikipedia.org/wiki/Vinogradov%27s_theorem (accessed June 27, 2009).

Von Mangoldt function. May 2, 2009. http://en.wikipedia.org/wiki/Von_Mangoldt_function (accessed June 30, 2009).

Wang, Yuan. The Goldbach Conjecture. London: World Scientific Publishing Company; 2 Sub Edition, 2003.

Weisstein, Eric W. Schnirelmann Constant. http://mathworld.wolfram.com/SchnirelmannConstant.html (accessed July 2, 2009).

—. Schnirelmann Density. http://mathworld.wolfram.com/SchnirelmannDensity.html (accessed July 2, 2009).

Education

Goldbach Conjecture