Mathematical mysteries: the Goldbach conjecture

Issue 2
May 1997


Prime numbers provide a rich source of speculative mathematical ideas. Some of the mystical atmosphere that surrounds them can be traced back to Pythagoras and his followers who formed secret brotherhoods in Greece, during the 5th Century BC. The Pythagoreans believed that numbers had spiritual properties. The discovery that some numbers such as the square root of 2 cannot be expressed exactly as the ratio of two whole numbers was so shocking to Pythagoras and his followers that they hushed up the proof!

Today, prime numbers are fascinating but they are also of commercial importance, since the best commercial and military ciphers depend on their properties. (See "Discovering new primes" in Issue 1 - it is yet to be proved that there are infinitely many Mersenne primes.)

Here is another unproved conjecture about prime numbers. It is called the Goldbach conjecture and may be stated as follows:

Definition of G

For example:

  • 8 = 3 + 5. Both 3 and 5 are prime numbers.
  • 20 = 13 + 7 = 17 + 3.
  • 42 = 23 + 19 = 29 + 13 = 31 + 11 = 37 + 5.

Notice that there can be more than one Goldbach pair. The conjecture says only that there is at least one, and has nothing to say about whether there may be more.

You can explore the Goldbach conjecture yourself with this Goldbach calculator. Simply enter an even integer, n, greater than 4 and the calculator will find all the Goldbach pairs. (The calculator does not account for Goldbach pairs for which the two primes are equal, such as 3+3=6 and 5+5=10.)

Historical Note

Christian Goldbach (1690-1764) was a Prussian amateur mathematician and historian who lived in St Petersburg and Moscow. He made his conjecture in a letter to Leonhard Euler, who at first treated the letter with some disdain, regarding the result as trivial. Goldbach's conjecture, however, remains unproved to this day.

Further reading

For an entertaining and revealing introduction to this problem, see Douglas R Hofstadter's book "Gödel, Escher, Bach" (Penguin Books 1979, ISBN 0 14 00 5579 7), especially the section "Aria with Diverse Variations" following chapter XII.

Comments

no result for 6.

hey why don't I see any result for 6?
Is it 1+5?

GOLDBACH'S CONJECTURE TI-84 PROGRAM

Any idea how to program the conjecture in a TI-84 calculator, yielding the pairs of odd primes given any even number greater than 4 and less than or equal to 4x10^18?

HOW TO CREATE THIS PROGRAM IN C#

how to create this kind of goldbach calculator in c# program ???

anyone can help me plz...

me too

if you find how to do in c++ pls tell me..

where is the calculator???

The calculator mysteriously disappeared from this page :(

odd meaning different or odd meaning 'not even'

Odd primes presumably means different primes. (rather than all the primes except 2, i.e. odd numbered primes)
Thus I fail to see how the number 6 (an even number greater than 4) can be partitioned into 2 odd primes. Unless of course you are considering the number 1 as a prime.
Of course 6 can be partitioned into 3 and 3 but then where does the 'odd prime' clause come in?
So I would word the conjecture in one of two ways -

"Every even number greater than 6 can be written as the sum of 2 different primes"
or
"Every even number greater than 4 can be written as sum of 2 primes which are not necessarily different"

Forgive me my ignorance... and thanks for the great article

Odd prime simply means any

Odd prime simply means any prime apart from 2. So in the case of 6 we have 6=3+3. Admittedly it's a strange way of formulating things, saying "odd prime" rather than "prime not equal to 2", but it's quicker to say!

aha - all primes except 2

Thanks for your reply, that's clear now.
My understanding is only very elementary, but these articles offer great insight.
Number 2, utterly simple yet so inexplicably awkward. Duality seems intrinsic to all of the universe and even contributes to the complexity of the GC. That is, we're proving a condition for a multiple of 2, concerning 2 numbers, neither of which will be multiple of 2.
Thanks again

GOLBACH'S CONJECTURE

Proven this year by a Peruvian mathematician, Harald Andrés Helfgott. He demonstrated this 271 year unsolved problem.
See link: http://www.truthiscool.com/prime-numbers-the-271-year-old-puzzle-resolved

GOLBACH'S CONJECTURE(S)

Helfgott proved a different conjecture. See above: "Every even number greater than 4 can be written as the sum of two odd prime numbers", versus the second sentence in the linked article: "H.A. Helfgott proved that any odd number greater than 5 can be written as the sum of 3 primes."

Lower bound in Goldbach's Conjecture

As shown in a recent paper (March 2013), see http://pubs.sciepub.com/ajma/1/1/2/abstract.html#.UYUcBEqSnRg, not only one but a minimum number of prime pairs (p,q) that fulfill the relationship 2n=p+q can be estimated using a very simple formula. Moreover, the centroids of triangles based on the triplets (6k-2,6k,6k+2) closely follow the average number of representations (also a simple formula).

Goldbach Conjecture

I have found an interesting result after looking at the Goldbach Conjecture.

It suggests that if there is a smaller Goldbach partition then there is always an equal or larger Goldbach partition.

Looks like a simple "proof" to me but I guess it may not be given this has eluded the mathematical world to date.

Anyway, I would like to get it reviewed by an expert in the field. It only uses two sides of A4.

Integers 2p, p prime

Hi. For some even integers - like 14, 38 and 122, for example - calculator didn't list all pairs ("7+7", "19+19" and "61+61" in these cases). I believe you have discard evident occurences such as integers 2p where p is prime.

Anonymous = "Alexandre"

Consideration in favor of Goldbach's Conjecture

Please read the following ArXiv paper on this topic:

http://arxiv.org/pdf/1208.2244.pdf

Is it a proof?

Goldbach's conjecture

I discovered a recent paper that offers a good literature overview on Goldbach's conjecture (2N=p+q, p,q=primes). It can downloaded at free from: http://vixra.org/pdf/1205.0108v1.pdf
Although it does not provide the complete mathematical proof, it shows many interesting numerical results such as an estimation of the pairs (p,q) that fufill the conjecture, which is shown to increase with N. I feel that this work may inspire some other researchers.

An alternative statement

The Goldbach pairs (Goldbach partitions), when considered multiplicatively rather than additively, can be used to generate what are known as odd semiprimes - the products of two odd prime factors (unique or identical). When you realize that fact, then you may make the conjecture that every perfect square integer can be expressed as the sum of an odd semiprime and another perfect square. We know that odd times odd will always generate odd, and we also know that the sequence of odds sum progressively to the sequence of perfect squares. We also know that the integers 0 and 1, the so-called identity elements, are also perfect squares. We also know that another way of stating the strong Goldbach conjecture is to say that every integer n is the average of two odd primes - hence, the Goldbach relates to midpoint. So we have the introduction of perfect squares and midpoint into the Goldbach conjecture which leads to a conclusion that the Goldbach conjecture has much deeper roots than the simple statement first suggested by Christian Goldbach. The conjecture relates to all the group axioms of arithmetic, along with the principles of elementary calculus and the principles of Euclidian geometry and the bilateral symmetry of the Cartesian grid.
I would suggest that the true gist of Goldbach lies in the principles of universality and reversibility that underlie the entire fabric of human logic and mathematics.
Example: Perfect square 25 = perfect square 4 plus semiprime 21=prime 3 x prime 7. Try a few billion others ( and don't forget those identity elements are also perfect squares)
And this all can be seen by using the group axioms and operations to see that every even integer can be rearranged as a model of the standard numberline. for example, the even integer 8:
0 1 2 3 4 5 6 7 8
8 7 6 5 4 3 2 1 0

Examine sums, products, differences of the binary columns along with the differences in the respective rows of those sums, products, and differences. The midpoint generates a perfect square and the Goldbach partitions generate semiprimes. And their differences are always (necessarily) another perfect square. When the Goldbach pairs are examined as the factors of semiprimes rather than as the addends of even integer sums and that those odd semiprimes are part of the sequential production of perfect squares - the Goldbach conjecture is far more tractable.
Play with it.

Mistake

This calculator does a great job, but does not list situations when the two prime numbers are the same (for example, 6 can only be expressed as 3+3, which does not calculate in the applet).

No mistake

The article explains this clearly - read the part in parenthesis.

The conjecture's wording

The conjecture is actually worded a bit differently.

Every even number greater than 2 can be written as the sum of two prime numbers.

The initial wording of the conjecture included 2 as a number that could be written as a sum of two prime numbers but that was also assuming 1 was a prime number.

4 is the first applicable number of the conjecture. It is also the only number than can have it's sums expressed with even numbers. 2 is the only even prime number. If you subtract 2 from an even number you get an even number which means it is at least divisible by 2, thus not a prime number.

Goldbach calculator

Dear Pluses and readers,

Finding that the calculator here: a) can't divide numbers like 4 and 6, and b) can't show all prime pairs for numbers like 26, I tried to reverse-engineer it, see why it can't do it, and suggest an improvement. Later, I found you have provided your algorithm (in detail) and it is its second part to blame.

I attach a link to my blog where one can see links to two algorithms of mine (not detailed), on the first of which a friend of mine based his calculator (there is a link to this calculator too). Please, don't be too critical, I'm neither a native English speaker, nor a mathemathician, nor a programmer.

Best regards,
Ivan

http://ivanianakiev.multiply.com/journal/item/599/Reverse_engineering_an...

wonderful it gets the deeper you dig

After receiving a B.S in Mathematics, i realized that i could just barely start to see the beauty in this marvelous subject...
Surprising it is that how simple conjectures can resist a solid proof for centuries...