Goldbach's conjecture

Letter from Goldbach to Euler dated June 7, 1742 (Latin-German)

The Goldbach Hypothesis, named after the mathematician Christian Goldbach , is an unproven statement from the field of number theory . As one of Hilbert's problems, it is one of the best-known unsolved problems in mathematics .

Strong (or binary) Goldbach conjecture

The strong (or binary) Goldbach Hypothesis is as follows:

Every even number that is greater than 2 is the sum of two prime numbers .

Many number theorists have dealt with this assumption up to the present day without having yet proven or disproved it.

By means of a volunteer computing project, Tomás Oliveira e Silva has meanwhile shown (as of April 2012) the validity of the assumption for all numbers up to 4 · 10 ¹⁸ . This is not proof that it holds for any arbitrarily large even number.

After the British publisher Faber & Faber offered a million dollar prize money to prove the presumption in 2000, public interest in this question also grew. The prize money was not paid because no evidence was received by April 2002.

Weak (or ternary) Goldbach hypothesis

The weaker guess

Any uneven number greater than 5 is the sum of three prime numbers.

is known as the ternary or weak Goldbach conjecture. It has been partially solved: On the one hand, it holds if the generalized Riemann Hypothesis is correct, and on the other hand, it has been shown that it holds for all sufficiently large numbers ( Winogradow's theorem , see related results ).

On May 13, 2013, the Peruvian mathematician Harald Helfgott announced a putative proof of Goldbach's ternary conjecture for all numbers greater than 10 ³⁰ . The validity for all numbers below 8.875 · 10 ³⁰ has already been checked with the aid of a computer.

From the strong Goldbach conjecture follows the weak Goldbach conjecture, because every odd number can be written as a sum . According to the strong Goldbach conjecture, the first summand is the sum of two prime numbers ( ), whereby a representation of as the sum of three prime numbers is found. ${\ displaystyle u}$ ${\ displaystyle u = (u-3) +3}$ ${\ displaystyle (u-3)}$ ${\ displaystyle u-3 = a + b}$ ${\ displaystyle u = a + b + 3}$ ${\ displaystyle u}$

Goldbach decompositions

Number of possibilities to represent the even numbers up to 200,000 as the sum of two prime numbers

A Goldbach decomposition is the representation of an even number as the sum of two prime numbers, for example a Goldbach decomposition of 8. The decompositions are not clear, as can be seen from. For larger even numbers there is a growing number of Goldbach decompositions (“multiple Goldbach numbers”). The number of Goldbach decompositions can be easily calculated with computer support, see illustration. ${\ displaystyle 3 + 5}$ ${\ displaystyle 18 = 7 + 11 = 5 + 13}$

In order to violate Goldbach's strong hypothesis, a data point would have to fall to the zero line at some point.

The requirement of an even number , that for each prime number with also a prime number, and thus a Goldbach decomposition is (the number that is the maximum number of Goldbach decompositions has) exactly fulfill the four numbers 10, 16, 36 and 210. Also the weaker requirement that for every prime number with also is a prime number does not meet any number . ${\ displaystyle n}$ ${\ displaystyle p}$ ${\ displaystyle n / 2 \ leq p <n}$ ${\ displaystyle np}$ ${\ displaystyle n = p + (np)}$ ${\ displaystyle n}$ ${\ displaystyle p}$ ${\ displaystyle n / 2 \ leq p <n-1}$ ${\ displaystyle np}$ ${\ displaystyle n> 210}$

Attempts to prove by amateurs

The Goldbach Hypothesis continues to attract amateur mathematicians. Occasionally, such attempts at proof also receive media attention.

Related results

In 1920 Viggo Brun proved that any sufficiently large even number can be represented as the sum of two numbers with a maximum of nine prime factors.
In 1930 Lew Genrichowitsch Schnirelman proved that every natural number is the sum of less than C prime numbers, where C is a constant that Schnirelman originally used to be 800,000 and which could later be reduced to 20.
In 1937, Ivan Matveevich Vinogradov proved that every odd number that is greater than a certain constant is the sum of three prime numbers (Vinogradov's theorem; weak Goldbach's conjecture for the case of sufficiently large numbers). Another proof of this was provided by Yuri Linnik in 1946 .
In 1937 Nikolai Grigoryevich Tschudakow proved that “almost all” even numbers are the sum of two prime numbers, that is, that the asymptotic density of the numbers that can be represented in this way is 1 in the even numbers.
In 1947 Alfréd Rényi proved that a constant K exists such that every even number is the sum of a prime number and a number with a maximum of K prime factors.
In 1966, Chen Jingrun proved that any sufficiently large even number is the sum of a prime number and a product of at most two prime numbers ( Chen's theorem ).
In 1995 Olivier Ramaré proved that every even number is the sum of at most six prime numbers.
In 2012, Terence Tao proved that any odd number greater than 1 is the sum of at most five prime numbers, thus improving Ramaré's result.

literature

Wolfgang Blum: Goldbach and the twins. In: Spectrum of Science Dossier 6/2009: The greatest riddles of mathematics. ISBN 978-3-941205-34-5 , pp. 34-39.
Apostolos Doxiadis : Uncle Petros and Goldbach's conjecture. Lübbe, 2000, ISBN 3-7857-0951-X ( fiction ).
Andrew Granville : Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis. In: Functiones et Approximatio 37. 2007, pp. 7-21 (English; PDF; 180 kB).
Melvyn B. Nathanson: Additive Number Theory. The Classical Bases. Springer-Verlag, New York 1996 (English).
Jörg Richstein: Verifying the Goldbach conjecture up to 4 · 10 ¹⁴ . In: Mathematics of Computation 70. 2001, pp. 1745-1749 (English).
Konstantin Fackeldey: The Goldbach Hypothesis and its previous attempts at a solution. Free University of Berlin (PDF).
Harald Helfgott: The ternary Goldbach problem , Oberwolfach Mathematical Snapshots , 2014

Web links

Goldbach conjecture verification project. By Tomás Oliveira e Silva.
Goldbach's Conjecture. Description in The Prime Glossary.
Goldbach. Splits an entered even number into two prime numbers.

Individual evidence

↑ In print in Paul Heinrich Fuss (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. (Volume 1), St.-Pétersbourg 1843, pp. 125-129.
↑ Jean-Marc Deshouillers , Gove Effinger, Herman te Riele , Dmitrii Zinoviev: A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electronic Research Announcements of the AMS 3, 1997, pp. 99-104 (English).
↑ Harald Andrés Helfgott: Minor Arcs for Goldbach's Problem. (PDF; 715 kB) and Major Arcs for Goldbach's Problem. (Preprint on arXiv .org; PDF; 1.1 MB)
↑ See Holger Dambeck: Weak Goldbach conjecture: solution for legendary number puzzle presented. On: SPIEGEL Online Science. May 23, 2013.
↑ Harald Andrés Helfgott, David J. Platt: Numerical Verification of the Ternary Goldbach Conjecture up to 8,875 10 ³⁰ . (Preprint on arXiv .org; PDF; 104 kB).
^ Jean-Marc Deshouillers , Andrew Granville , Władysław Narkiewicz , Carl Pomerance : An upper bound in Goldbach's problem. Mathematics of Computation 61, No. 203, July 1993, pp. 209-213.
↑ Holger Dambeck: Hobby Mathematician: Ingenious proof, but unfortunately wrong. On: Spiegel.de. December 17, 2013, accessed August 23, 2014.
↑ Juri Linnik: On the eighth Hilbert problem. In: Pavel S. Alexandrov (ed.): The Hilbert problems. Harri Deutsch, 1998.
↑ Chen Jingrun : On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao 17, 1966, pp. 385-386 (Chinese); Scientia Sinica 16, 1973, pp. 157-176 (English; Zentralblatt review ); Scientia Sinica 21, 1978, pp. 421-430 (English; Zentralblatt review ).
↑ Olivier Ramaré: On Šnirel'man's constant. Annali della Scuola Normale Superiore di Pisa 22, 1995, pp. 645-706 (English).
^ Terence Tao : Every odd number greater than 1 is the sum of at most five primes. Mathematics of Computation (English; arxiv : 1201.6656 ).

[1] In print in Paul Heinrich Fuss (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. (Volume 1), St.-Pétersbourg 1843, pp. 125-129.

[2] Jean-Marc Deshouillers , Gove Effinger, Herman te Riele , Dmitrii Zinoviev: A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electronic Research Announcements of the AMS 3, 1997, pp. 99-104 (English).

[3] Harald Andrés Helfgott: Minor Arcs for Goldbach's Problem. (PDF; 715 kB) and Major Arcs for Goldbach's Problem. (Preprint on arXiv .org; PDF; 1.1 MB)

[4] See Holger Dambeck: Weak Goldbach conjecture: solution for legendary number puzzle presented. On: SPIEGEL Online Science. May 23, 2013.

[5] Harald Andrés Helfgott, David J. Platt: Numerical Verification of the Ternary Goldbach Conjecture up to 8,875 10 ³⁰ . (Preprint on arXiv .org; PDF; 104 kB).

[6] Jean-Marc Deshouillers , Andrew Granville , Władysław Narkiewicz , Carl Pomerance : An upper bound in Goldbach's problem. Mathematics of Computation 61, No. 203, July 1993, pp. 209-213.

[7] Holger Dambeck: Hobby Mathematician: Ingenious proof, but unfortunately wrong. On: Spiegel.de. December 17, 2013, accessed August 23, 2014.

[8] Juri Linnik: On the eighth Hilbert problem. In: Pavel S. Alexandrov (ed.): The Hilbert problems. Harri Deutsch, 1998.

[9] Chen Jingrun : On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao 17, 1966, pp. 385-386 (Chinese); Scientia Sinica 16, 1973, pp. 157-176 (English; Zentralblatt review ); Scientia Sinica 21, 1978, pp. 421-430 (English; Zentralblatt review ).

[10] Olivier Ramaré: On Šnirel'man's constant. Annali della Scuola Normale Superiore di Pisa 22, 1995, pp. 645-706 (English).

[11] Terence Tao : Every odd number greater than 1 is the sum of at most five primes. Mathematics of Computation (English; arxiv : 1201.6656 ).