Full resolution (TIFF) - On this page / på denna sida - Sidor ...
<< prev. page << föreg. sida << >> nästa sida >> next page >>
Below is the raw OCR text
from the above scanned image.
Do you see an error? Proofread the page now!
Här nedan syns maskintolkade texten från faksimilbilden ovan.
Ser du något fel? Korrekturläs sidan nu!
This page has never been proofread. / Denna sida har aldrig korrekturlästs.
Redaktionen er Professor Hardy meg’et taknemlig for den
Venlighed, hvormed han har imødekommet vor Opfordring til at
offentliggøre sit i Matematisk Forening holdte Foredrag i
Matematisk Tidsskrift og derved givet Tidsskriftets Læsere en enestaaende
Lejlighed til at faa et Indblik i de dybtgaaende nye
Undersøgelsesmetoder, hvormed Professor Hardy i Samarbejde med Professor
Littlewood har beriget Talteorien, og ved hvilke det er lykkedes
dem at angribe og løse klassiske Opgaver, der hidtil var betragtet
som ganske utilgængelige.
Goldbach’s Theorem.
By G. H. Hardy1).
The famous problem of which I propose to speak tonight
is probably as difficult as any of the unsolved problems of
mathematics, and my lecture cannot be entirely easy. I must
be content if I can give you some rough notion of the nature
and of the history of the problem, and of the general ideas
which have guided the mathematicians of the past and of the
present in their efforts to find a solution.
Every even number is the sum of two primes:
this is ’Goldbach’s Theorem’. I may perhaps begin by the
trivial observation that, if ’prime’ is to mean what it means in
modern mathematics, the theorem is obviously false. It fails
for 2, which is a prime, but not the sum of two. We do not
nowadays call i a prime, for, if we do, the factorisation of
a number into primes is not unique. We must therefore insert
the words ’greater than 2’ in the enunciation of the
theorem.
The theorem is stated in Goldbach’s correspondense with
Euler in the year 1742. It would seem that he had been
anticipated by Descartes2). It was conjectured independently
by Waring a little later, and it is, I believe, in Waring’s
Meditationes algebraicae (1770) that the conjecture
appears first in print. Each of these authors appears to have
observed that, if every even number (greater than 2) is the
sum of two primes, then every number (greater than 5) is
*) A lecture to the Mathematical Society of Copenhagen on O.October 1921.
2) In these matters of history I am content to follow Prof. L. E. Dicksons
History of the theory of numbers (Washington, 1919, vol. I. pp. 421
et seq.). Dickson however attributes to Descartes the assertion that
’every even number is a sum of I, 2, or 3 primes’, and here the word
’even’ should surely be deleted.
Mat. Tidsskr. B. 1922, I
<< prev. page << föreg. sida << >> nästa sida >> next page >>
