Goldbach's weak conjecture
Encyclopedia
In number theory
, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that:
This conjecture
is called "weak" because if Goldbach's strong conjecture
(concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.)
The conjecture has not yet been proven, but there have been some useful near misses. In 1923, Hardy
and Littlewood
showed that, assuming the generalized Riemann hypothesis
, the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem
) that all sufficiently large odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective Siegel–Walfisz theorem, did not give a bound for "sufficiently large", his student K. Borozdin proved, in 1939, that 314348907 is large enough. This number has 6,846,169 decimal digits, so checking every number under this figure would be highly infeasible with current technology.
In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered this threshold to approximately
. The exponent is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 
for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.) However, this bound is small enough that any single odd number below the bound can be verified by existing primality test
s such as elliptic curve primality proving
, which generates a proof of primality and has been used on numbers with as many as 26,643 digits.
In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed that the generalized Riemann hypothesis
implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 1020 with an extensive computer search of the small cases.
Leszek Kaniecki showed every odd integer is a sum of at most five primes, under Riemann Hypothesis
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that:
- Every odd number greater than 7 can be expressed as the sum of three odd primesPrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. (A prime may be used more than once in the same sum.)
This conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
is called "weak" because if Goldbach's strong conjecture
Goldbach's conjecture
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...
(concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.)
The conjecture has not yet been proven, but there have been some useful near misses. In 1923, Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
and Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
showed that, assuming the generalized Riemann hypothesis
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
, the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem
Vinogradov's theorem
In number theory, Vinogradov's theorem implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after...
) that all sufficiently large odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective Siegel–Walfisz theorem, did not give a bound for "sufficiently large", his student K. Borozdin proved, in 1939, that 314348907 is large enough. This number has 6,846,169 decimal digits, so checking every number under this figure would be highly infeasible with current technology.
In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered this threshold to approximately
. The exponent is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.) However, this bound is small enough that any single odd number below the bound can be verified by existing primality testPrimality test
A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not...
s such as elliptic curve primality proving
Elliptic curve primality proving
Elliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number . It is a general-purpose algorithm, meaning it does not depend on the number being of a special form...
, which generates a proof of primality and has been used on numbers with as many as 26,643 digits.
In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed that the generalized Riemann hypothesis
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 1020 with an extensive computer search of the small cases.
Leszek Kaniecki showed every odd integer is a sum of at most five primes, under Riemann Hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
.