## Historical Context
Euler is generally known as one of the g...

The first series has denominators of the form 4$k$-1, where 4$k$-1 ...

This conjecture is one of the major unsolved problems in number the...

It is just in the last paragraph that Goldbach mentions his conject...

It was only more than a century later, in 1856, that Moritz Stern,a...

After Stern's find there was a question left unanswered: is there a...

A Not So Famous Goldbach Conjecture

Goldbach to Euler

Petersburg, November 18th, 1752

Sir,

you kindly wrote me some time ago that the books

which Mr. Spener addressed to me were already dis-

patched from Berlin in July; however, up to this moment

I do not know to which merchant in Petersburg they were

addressed or if they were even left behind at Lübeck. As

it is not known to me by what shortcuts you obtained

the diﬀerence of the series

1

3

+

1

7

+

1

11

+

1

19

+ . . . and

1

5

+

1

13

+

1

17

+

1

29

+ . . . ,if you have already published the

method, please tell me where it is to be found. I take it to

be certain that the number of all primes of the form a

2

+1

is inﬁnitely large, even if I cannot immediately prove it,

and I do not think your reason for doubting it - that the

number of primes of the form a

2

+ 1 is inﬁnitely smaller

than the number of all primes - is at all relevant, since

no inﬁnite number can be taken so small that it is not

inﬁnitely greater than some other inﬁnite. Mersenne as-

suredly did not say that only ten perfect numbers are pos-

sible, as he himself indicates eleven, among which your

eighth one, Sir, is however not comprised; he does not

state either that the number of perfect numbers is ﬁnite,

but only that no range of numbers can be indicated which

is so large that it could not be devoid of perfect numbers.

You will be able to see that for yourself - in case tome

II of the Commentarii has still not arrived - from the

general preface to Mersenne’s Cogitata Physico Mathe-

matica, §19, as cited by Mr.Winsheim. As regards Le-

uneschloß’s Paradoxa mathematica, they were printed at

Heidelberg in 1658, in octavo; however, I never possessed

this book myself, but borrowed the copy which I read in

1716 from the Old City Library at Königsberg, and re-

turned it there before I departed for the last time in 1718,

so it is impossible that you could have seen the book at

my place in Petersburg. On the other hand, I remember

- though not with utter certainty - that some years ago

you wrote me from Berlin you had got the book, along

with Bongo’s 1591 list, from the Royal Library. I am not

aware of anything further about this Leuneschloß than

that I read somewhere he had been a professor at Heidel-

berg; if I am not mistaken, he is also referred to by some

as ’Luneschlos’, so his name should have to be looked up

in the encyclopedias in both spellings. It seems that he

found his paradox on perfect numbers in Mersenne and

afterwards, when writing it down from memory, deviated

from Mersenne’s true meaning. I had already observed

in one of my earlier letters that no algebraic formula can

yield only prime numbers; indeed, taking the formula to

be, for example, x

3

+ bx

2

+ cx + e, it is obvious that

whenever x is a multiple of the absolute term e, then

(and consequently inﬁnitely often) the formula will yield

a non-prime number; but if e were to equal 1, I merely

substitute x+p for x, as then the formula is changed into

x

3

+ 3px

2

+ 3p

2

x + p

3

+bx

2

+ 2bpx + bp

2

+cx + cp

+ 1

(1)

and whenever x is a multiple of the number p

3

+ bp

2

+

cp + 1, the formula yields a non-prime number. Now

since this case where the highest power of x equals 3 can

be extended to all other whims of nature, whatever the

power of x is, it is impossible to indicate an algebraic

series in which there should not occur inﬁnitely many

terms consisting of non-prime numbers. I have yet an-

other small theorem to add, which is quite new and which

I take to be true until the contrary is proved: Any odd

number equals the sum of twice a square and a prime,

or: 2n − 1 = 2a

2

+ p, where a is to signify a whole num-

ber, including 0, and p a prime number; for example,

17 = 2 · 0

2

+ 17; 21 = 2 · 1

2

+ 19; 27 = 2 · 2

2

+ 19, and

so on. With a dutiful recommendation to your dearest

family, I remain, Sir, your most devoted servant

Goldbach.

The portrait on the right is certainly no Goldbach. It's evident from his dress first and foremost. All the arguments are well summarized here: https://hsm.stackexchange.com/a/559/8163
The internet for some reason thinks that it's Goldbach.
This conjecture is one of the major unsolved problems in number theory, being one of the four problems that [Landau in 1912](https://en.wikipedia.org/w/index.php?title=Landau%27s_problems&oldid=869848981) described as “unattackable at the present state of mathematics”. (All four problems [remain open](https://terrytao.wordpress.com/2014/11/13/254a-announcement-analytic-prime-number-theory/).)
Goldbach is right here about Euler's intuition being mistaken: the fact that the density of numbers of the form $a^2 + 1$ is less than that of the prime numbers is not relevant to the question of whether there are infinitely many of them.
The first series has denominators of the form 4$k$-1, where 4$k$-1 isn't prime. The second series has denominators of the form 4$k$+1, where 4$k$+1 isn't prime.
Where $4k-1$ (resp. $4k+1$) is prime?
It is just in the last paragraph that Goldbach mentions his conjecture: every odd integer can be written in the form $p + 2a^2$, where $p$ is a prime (or 1, then considered a pime) and $a\geq0$ is an integer. In his reply in December that year, Euler wrote back saying that he had verified this up to 1000 and later he extended the verification up to 2500. Euler also mentioned that as the odd number $2n + 1$ increased, the number of representations tended to increase.
It was only more than a century later, in 1856, that Moritz Stern,a mathematician at the university of Gottingen, was able to test the conjecture up to 9000 with the help of his students. At the time, a lot of primality testing (checking if a number is prime) was done with exhaustive divisibilty testing ([sieve of Erastothenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes)) done by hand, thus taking a long time. Stern was able to find 2 exceptions: 5777 and 5993, that to this day are the only known exceptions to Goldbach's conjecture.
It's interesting to note that with today's available computing power anyone can easily find the exceptions in a fraction of a second using a regular computer. Below is the code you can use to find the exceptions:
```C
private bool isTwiceSquare(long number) {
double squareTest = Math.Sqrt(number/2);
return squareTest == ((int)squareTest);
}
```
```C
int[] primeList = ESieve(10000);
int result = 1;
bool notFound = true;
while(notFound){
result += 2;
int j = 0;
notFound = false;
while (result >= primeList[j]) {
if(isTwiceSquare(result-primeList[j])){
notFound = true;
break;
}
j++;
}
}
```
Complete source code [here](https://www.mathblog.dk/files/euler/Problem46.cs)
## Historical Context
Euler is generally known as one of the greatest mathematicians of his time, with a vast body of work and extended contributions in math and physics. Christian Goldbach, on the other hand, is known by most people because of one single idea: the Goldbach Conjecture - a statement in number theory which is
very easy to understand – every even integer greater than 2 is the sum of two prime numbers - but still unproved more than 250 years after he first wrote it down in 1742.
![](https://i.imgur.com/tdG9IId.jpg)
Goldbach was born in Germany in 1690, 17 years before Euler, and was often unfairly described as an "amateur" mathematician or a mathematician who chose to pursue a professional career in law. The relationship between Euler and Goldbach begins in 1727 in St. Petersburg: Euler, 20 years old at the time, had just gotten a position in the Petersburg Academy and was eager to prove his talent, while 37 year old Goldbach was responsible for supervising the young Tsar's education. Since then, Euler and Goldbach corresponded frequently through letters almost until Goldbach's death in 1764 (196 letters in total).
These letters constitute an incredible opportunity to delve into Euler and Goldbach's lives: from Euler's rise in the mathematical world to Goldbach's explorations that lead to several of his theorems and conjectures,as well as their own personal and professional struggles (if you want to read the letters they have been translated to English and edited in this [book](https://www.springer.com/gp/book/9783034808927)). The letter you are reading now was written by Goldbach in 1752 (after his most famous conjecture), where Goldbach expressed another of his conjectures: he believed that every positive odd integer is either prime or the sum of a prime and twice a square. In the following comments we are going to test the conjecture and explore the history that came after.
If you want to read the original version of the letter in German you can have access to it [here](http://eulerarchive.maa.org/correspondence/letters/OO0878.pdf).
After Stern's find there was a question left unanswered: is there a finite number of exceptions to Goldbach's conjecture?
No one has been able to prove this but there are a few things that might indicate that the number of exceptions is indeed finite, one of them being the fact that the average number of ways of expressing an odd integer $2n - $1 as a prime plus twice a square increases without limit as $n$ increases.
Actually it is not difficult to see that if the number of primes up to $n$ is on the order of $\frac{n}{\log n}$ while
the number of squares up to $n$ is on the order of $n^{1/2}$, the average number of ways of
expressing odd numbers up to $2n-1$ as a prime plus twice a square is at least
$$
n^{1/2}\frac{n}{\log n}\frac{1}{2n-1} \approx \frac{n^{1/2}}{\log n}
$$