#9
@CalbernandHowbe every prime number (except 2) is odd. we also know that odd + odd is always even.
for the statement: "Is every even number above 2 the sum of exactly 2 primes?"
answer is yes.
e.g., 7+5 = 12; 11+19 = 30; 3+3 = 6
for the statement: "Is every even number above 2 the sum of exactly 2 primes?"
answer is yes.
e.g., 7+5 = 12; 11+19 = 30; 3+3 = 6
My dad asked me whether every integer is the sum of three cubes...
@AOOP09 how do you write the following numbers as the sum of three primes? 3, 4, 5, 6, 8,... ?
@AOOP09 how do you write the following numbers as the sum of three primes? 3, 4, 5, 6, 8,... ?
@tourdivoire question in #3 said "2 primes" not 3.
<Comment deleted by user>
@tourdivoire Goldbach's Conjecture.Here is what wikipedia says:
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII),[4] in which he proposed the following conjecture:
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until all terms are units.
Goldbach was following the now-abandoned convention of considering 1 to be a prime number,[2] so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which is easily seen to imply the first:
Every integer greater than 2 can be written as the sum of three primes.[5]
Euler replied in a letter dated 30 June 1742[6] and reminded Goldbach of an earlier conversation they had had ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
Every positive even integer can be written as the sum of two primes.
This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated:[7][8]
"Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." ("That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.")
Each of the three conjectures above has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is:
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until either all terms are two (if the integer is even) or one term is three and all other terms are two (if the integer is odd).
A modern version of the marginal conjecture is:
Every integer greater than 5 can be written as the sum of three primes.
And a modern version of Goldbach's older conjecture of which Euler reminded him is:
Every even integer greater than 2 can be written as the sum of two primes.
These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer {\displaystyle N=p+1}{\displaystyle N=p+1} larger than 4, for {\displaystyle p}p a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version (without of course being a counterexample to the original version) of the third conjecture. The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer {\displaystyle n}n, could not possibly rule out the existence of such a specific counterexample {\displaystyle N}N). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture," asserts that
Every odd integer greater than 7 can be written as the sum of three odd primes,
A proof for the weak conjecture was proposed in 2013 by Harald Helfgott. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since.[9][10][11] Note that the weak conjecture would be a corollary of the strong conjecture: if n – 3 is a sum of two primes, then n is a sum of three primes. But the converse implication and thus the strong Goldbach conjecture remain unproven.
Verified results
For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.[12] With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 1018 (and double-checked up to 4 × 1017) as of 2013. One record from this search is that 3325581707333960528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII),[4] in which he proposed the following conjecture:
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until all terms are units.
Goldbach was following the now-abandoned convention of considering 1 to be a prime number,[2] so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which is easily seen to imply the first:
Every integer greater than 2 can be written as the sum of three primes.[5]
Euler replied in a letter dated 30 June 1742[6] and reminded Goldbach of an earlier conversation they had had ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
Every positive even integer can be written as the sum of two primes.
This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated:[7][8]
"Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." ("That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.")
Each of the three conjectures above has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is:
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until either all terms are two (if the integer is even) or one term is three and all other terms are two (if the integer is odd).
A modern version of the marginal conjecture is:
Every integer greater than 5 can be written as the sum of three primes.
And a modern version of Goldbach's older conjecture of which Euler reminded him is:
Every even integer greater than 2 can be written as the sum of two primes.
These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer {\displaystyle N=p+1}{\displaystyle N=p+1} larger than 4, for {\displaystyle p}p a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version (without of course being a counterexample to the original version) of the third conjecture. The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer {\displaystyle n}n, could not possibly rule out the existence of such a specific counterexample {\displaystyle N}N). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture," asserts that
Every odd integer greater than 7 can be written as the sum of three odd primes,
A proof for the weak conjecture was proposed in 2013 by Harald Helfgott. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since.[9][10][11] Note that the weak conjecture would be a corollary of the strong conjecture: if n – 3 is a sum of two primes, then n is a sum of three primes. But the converse implication and thus the strong Goldbach conjecture remain unproven.
Verified results
For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.[12] With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 1018 (and double-checked up to 4 × 1017) as of 2013. One record from this search is that 3325581707333960528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.
@AOOP09 wrote "Every integer greater than 2 can be written as the sum of three primes."
#17
Nope, read #16
Nope, read #16
@tourdivoire He considered 1 as a prime:
Christian Goldbach wrote a letter to Leonhard Euler proposing him the conjecture that every integer greater than one can be written as the sum of at most three primes (CONSIDERING 1 AS A PRIME)
After Euler’s reply, Goldbach stated a new conjecture: every even integer greater than two can be written as the sum of two primes.
This is the famous Goldbach conjecture that has evaded mathematicians’ efforts since its statement.
Anybody who has a correct proof of it or a counterexample would instantly become famous. The net is full of claimed proofs: none has survived serious scrutiny; most are utter nonsense.
It has been verified for all even numbers greater than two and less than 4⋅1017 , but of course this is not a proof.
However, it has been proved that every even integer greater than two is the sum of at most four primes. Whether this is a step towards a proof of the Goldbach conjecture is not clear. See Goldbach's conjecture - Wikipedia for more information and references.
Christian Goldbach wrote a letter to Leonhard Euler proposing him the conjecture that every integer greater than one can be written as the sum of at most three primes (CONSIDERING 1 AS A PRIME)
After Euler’s reply, Goldbach stated a new conjecture: every even integer greater than two can be written as the sum of two primes.
This is the famous Goldbach conjecture that has evaded mathematicians’ efforts since its statement.
Anybody who has a correct proof of it or a counterexample would instantly become famous. The net is full of claimed proofs: none has survived serious scrutiny; most are utter nonsense.
It has been verified for all even numbers greater than two and less than 4⋅1017 , but of course this is not a proof.
However, it has been proved that every even integer greater than two is the sum of at most four primes. Whether this is a step towards a proof of the Goldbach conjecture is not clear. See Goldbach's conjecture - Wikipedia for more information and references.
@AOOP09 I know Goldbach conjecture for fox sake. Now read your own writings and write 6 as the sum of three primes please. lichess.org/forum/off-topic-discussion/i-can-help-you#5
This topic has been archived and can no longer be replied to.