Translation of Goldbach's 1742 letter to Euler - MathOverflow

Translation of Goldbach's 1742 letter to Euler

2010-05-03T15:57:38Z

This ought to be a simple one to answer. Does anyone know of, or can anyone provide, an accurate English translation of the marginal remarks in Goldbach's letter to Euler

http://upload.wikimedia.org/wikipedia/commons/1/18/Letter_Goldbaxh-Euler.jpg

in which a statement equivalent to the Goldbach conjecture is first stated?

Answer by vonjd for Translation of Goldbach's 1742 letter to Euler

2010-05-03T16:31:09Z

One of the problems seems to be the handwriting - at least here is a copy which is transcribed, perhaps this helps to get the gist (the German/Latin is even for today's native speakers hard to understand):

http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0765.pdf

Answer by Homology for Translation of Goldbach's 1742 letter to Euler

2010-05-03T18:06:25Z

I'm not really a language expert, but I think the last sentence of the footnote reads: "At least it seems that every number greater than 1 is an 'aggregatum trium numerorum primorum'" and 'aggregatum trium numerorum primorum' should mean "sum of three primes" (and it should really mean "sum of three or less primes")

Edit: correction, thanks to José

Answer by Bertram Arnold for Translation of Goldbach's 1742 letter to Euler

2010-05-03T19:13:46Z

I do not have the reputation to comment on the answer, so I have to start a new one.

Freely translating page 127 of http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0765.pdf :

I deem it to be advantageous to note the following conjecture, even though it lacks a proof, as a counterexample could provide further insights. Fermat's idea that every number of the form $2^{2^{n-1}}+1$ is prime can, as you have shown, not be true; but it would be strange if this series yielded a lot of "numeros unico modo in in duo quadratis divisibiles" (numbers that can be divided into two squares???). I, too, would like to hazard a conjecture: that every number that is the sum of two primes is the sum of arbitrary numbers of primes (or 1), except the "congierem omnium unitatum" (the collections of all in one???; the footnote was already translated by Mark), for example...

Answer by Franz Lemmermeyer for Translation of Goldbach's 1742 letter to Euler

2010-05-03T19:15:09Z

I have an English translation of the letter; you can find my email address on my homepage. Here is the relevant part:

"By the way, I take it to be not useless to note down also such propositions that are very probable, even if a real proof is lacking; for even if afterwards they were found to be erroneous, they could all the same give occasion for the discovery of a new truth. Thus Fermat's idea that all the numbers $2^{2^{n-1}}+1$ yield a series of prime numbers cannot hold up, as you already demonstrated, Sir; but it should still be remarkable if this series were composed only of numbers that could be split into two squares in a unique way. I should like to risk another conjecture of that kind: any number composed from two primes is the sum of as many prime numbers (including $1$) as one wishes, right down to the sum that consists just of ones.

After reading this through again, I see that the conjecture can be proved quite rigorously for the case $n+1$ if it holds for the case $n$ and $n+1$ can be split into two prime numbers. The proof is very easy; and at least it appears to be true that every number greater than $2$ is the sum of three prime numbers."

The translation was done by Martin Mattmüller.