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 Prashar
Word count: 3718 words
Abstract 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.

2


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

3


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:

6 = 3+3
8 = 3+5
10 = 3 + 7 = 5 + 5 Part 2 states that 9, 11, 13, 15 etc. (all odd integers) can be
12 = 5 + 7
displayed as the sum of three primes. Examples are shown below:

9 = 2+ 2+5 = 3+3+3
11 = 2 + 2 + 7 = 3 + 3 + 5
13 = 3 + 3 + 7
15 = 2 + 2 + 11 = 3 + 5 + 7

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.

II) The Binary and Ternary Conjectures; Linked

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

4


Let N = a sufficiently large positive, even integer
p1 and p2 = two prime numbers (may be identical)

If N = p1 + p 2
(i.e. If Goldbach’s binary conjecture is true)

Then
N + 3 = p1 + p 2 + 3
∴ N + 3 = p1 + p 2 + p 3 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 O = p1 + p 2 + p 3
(i.e. if Goldbach’s ternary conjecture is true)

Then
O + 3 = p1 + p 2 + p 3 + 3
∴ O + 3 = p1 + p 2 + p 3 + p 4

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

5


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].

6


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:

8 = 3+5
18 = 5 + 13 = 7 + 11
24 = 5 + 19 = 7 + 17 = 11 + 13

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 1.2 × 1018 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
probability Probability
Interval % prime Matching Interval % prime
prime pair neither prime pair
0 - 10 60 90 – 100 20 0.12 0.88
10 - 20 80 80 – 90 40 0.32 0.68
20 - 30 40 70 – 80 60 0.24 0.76
30 - 40 40 60 – 70 40 0.16 0.84
40 - 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).

7


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.

8


‘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 s0 ≥ s such
that every integer > 1 is a sum of at most s primes. The smallest proven value of s 0
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:

‘ σ ( A ⊕ B) ≥ min{1,σ ( A) + σ ( B )} ’7
• σ represents the Schnirelmann density, defined as the ‘greatest lower bound of the

S ( n)
fractions where S (n) is the number of terms in the set ≤ n ’8.
n
• {n ≥ 1, n ∈ Q} where Q represents a set.
e.g. Set: {0, 2, 4, 6, 8}
5
When n = 8 , S (n) = 5 {σ < }
8
In this situation, trial and error was used to calculate the Schnirelmann density; the

S ( n)
greatest lower bound is the smallest value of which in this case was < 5 8 .
n
• Greatest Lower Bound: If it is given the value c in set Q , then c ≤ x, ∀x ∈ Q . This
could otherwise be seen as 0 ≤ c ≤ smallest value of x .
• A ⊕ B = {a + b, a ∈ A, b ∈ B} represents the direct sum, where each element of set
A is added to every element of set B, any repeated values are discarded.
• min( A, B ) represents the smallest value in the set A or B, depending on which one
is smaller.

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).

9


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
bound of set B. In terms of Schnirelmann’s theorem, one very important point to
remember is that 0 ∈ A ∩ B , as mentioned earlier.

Let P = {0,1} ∪ { p} = {0,1, 2,3, 5, 7, ...}
Where p is all prime numbers
Q= P+P

= {0,1,2,3,5,7,1,2,3,4,6,8,2,3,4,5,7,9,3,4,5,6,8,10,5,6,7,8,10,12,7,8,9,10,12,14}
By using the direct sum method and discarding all repeated values, you get:
Q = {0,1,2,3,4,5,6,7,8,9,10,12,14...}

It can be shown using the inclusion-exclusion principle9 that:
σ ( P) = 0
But σ (Q) > 0

S ( n)
Now if we consider that σ (Q) is the greatest lower bound of , and is > 0 , then the
n
following method applies:

P = {0,1, 3, 5, 7,11...} σ ( P) = 0
Q = {0,1, 2, 3, 4, 5,6,8,10,12,14...} σ (Q) > 0

Remember that as σ ( P + P) ≥ min{1,σ ( P ) + σ ( P )} , the maximum value that
σ ( P + P ) can ever take, depending on how many times P is added, will be 1.

Therefore if this process is then repeated using Q + Q , then Q + Q + Q + Q , this leads
to:
σ (kQ) = kσ (Q) = 1 (This is considered an acceptable rule)

9
Inclusion-Exclusion Principle. June 26, 2009. http://en.wikipedia.org/wiki/Inclusion-
exclusion_principle (accessed July 1, 2009).

10


1 10
∴k =
σ (Q)
This is because eventually S (n) = n , 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

S (n)
if the set contains all positive integers, because then = 1 at every value.
n
Therefore if the process is repeated k 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

15
Deshouillers (1973)
9

11
Klimov et al. (1972)
5

55 Klimov (1975)

27 Vaughan (1977)

26 Deshouillers (1977)

19 Riesel and Vaughan (1983)

7 Ramaré (1995)

10
Schnirelmann's Theorem. June 27, 2009.
http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem (accessed July 1,
2009).
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).

11


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
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) / n
6 3 3 0.500000000
12 5 7 0.416666667
30 7 23 0.233333333
98 19 79 0.193877551
220 23 197 0.104545455
308 31 277 0.100649351
556 47 509 0.084532374
992 73 919 0.073588710
2642 103 2539 0.038985617
5372 139 5233 0.025874907
7426 173 7253 0.023296526
43532 211 43321 0.004847009
54244 233 54011 0.004295406
63274 293 62981 0.004630654
113672 313 113359 0.002753536
128168 331 127837 0.002582548
194428 359 194069 0.001846442
194470 383 194087 0.001969455
413572 389 413183 0.000940586
503222 523 502699 0.001039303
1077422 601 1076821 0.000557813
3526958 727 3526231 0.000206127
3807404 751 3806653 0.000197247
10759922 829 10759093 0.000077045
24106882 929 24105953 0.000038537
27789878 997 27788881 0.000035876
37998938 1039 37997899 0.000027343
113632822 1163 113631659 0.000010235
187852862 1321 187851541 0.000007032

13
(Herkommer 2004)

12


335070838 1427 335069411 0.000004259
419911924 1583 419910341 0.000003770
721013438 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.

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)

14
ibid.

13


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:

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.

14


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
ln ln x 2
π 2 ( x ) = O( x ( ))
ln x

• π 2( x) is defined as { p = x p + 2 = q} , where p and q are both prime. The

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

15


• The ‘O’ function is the Landau notation. This can be described as follows:
As x → ∞
f ( x) = O( g ( x))
But this is only if for a large value of x there exists positive real integers for M
and x 0 such that:
f ( x) ≤ M g ( x) For all values of x

Where M is a constant19

This basically means that the Landau notation is used to help express f (x ) as x

tends to infinity in the form of a simplified g (x) .

For example:
Let f ( x) = 3 x 3 − 4 x 2 + 7

Now we want to consider the behaviour of f (x ) as x → ∞ . It is obvious that 3x 3

will grow the fastest, as it has the highest derivative ( 9x 2 ).

f ( x) = O( x 3 ) [Remember x → ∞ ]

Now this needs to be in the same form as f ( x) ≤ M g ( x) , where M and x 0 are
positive, real integers.
Let x0 = 1

3 x 3 − 4 x 2 + 7 ≤ 3x 3 + 4 x 2 + 7 [Modulus has been taken]

3 x 3 − 4 x 2 + 7 ≤ 3x 3 + 4 x 3 + 7 x 3 [ x0 = 1 ]

3 x 3 − 4 x 2 + 7 ≤ 14 x 3 [Collect like terms]

Thus to use the original form:
3 x 3 − 4 x 2 + 7 ≤ 14 x 3

For all x > x 0

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

16


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.

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. 3 + 5 = 8, 5 + 7 = 12 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:

1
∏ (1 − ( p − 1)
p >2
2
)

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
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).

17


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) N = p1 + p 2
2) N = p1 + p 2 p 3
(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.

22
PrimeFan. Chen's Theorem. http://planetmath.org/encyclopedia/ChensTheorem.html (accessed June
27, 2009).

18


VII) Graphical methods and Vinogradov’s Theorem
An interesting method pursued by some mathematicians is the use of graphs23:

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

23
(Herkommer 2004)

19


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
N → ∞ . This is a quantity that is required in order to assess r(N) asymptotically
(as it tends to infinity).

And:
r(N ) = ∑ [Λ( p1) Λ( p2) Λ( p3)]
p1+ p 2 + p 3= N

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

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

Therefore:

r(N ) = ∑ [Λ ( p1) Λ ( p2) Λ ( p3)]
p1+ p 2 + p 3= N

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

20


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:

N 2 log −3 N ≤ 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 1.2 × 10 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).

21


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

22


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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.
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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

Schnirelmann's Theorem. June 27, 2009.
http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem

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).

23


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

24

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