GOLDBACH’SCONJECTURE

Simple, but Unproved

Goldbach’s Conjecture

Christian Goldbach, March 18, 1690 - November 20, 1764, stated that:

“Every even number greater than two can be written as the sum of two primes”

(the same prime may be used twice)

Goldbach’s Conjecture

• One of the oldest unsolved problems in number theory and in all of mathematics.

• The conjecture is divided into two parts known as the weak and strong conjecture respectively.

• The conjecture has had additions since Goldbach’s original theory was presented.

Goldbach’s Conjecture

• Goldbach originally described his conjectures in a letter to Leonhard Euler in 1742.

• “every number greater than five can be written as the sum of three primes” (weak)

• “every even number greater than two can be written as the sum of two primes” (strong)

Examples• 4 = 2 + 2• 6 = 3 + 3• 8 = 3 + 5• 10 = 3 + 7 and 5 + 5• 12 = 5 + 7• 14 = 3 + 11 and 7 + 7

• 54 = 7 + 47 and 11 + 43 and 13 + 41 and 17 + 37 and 23 + 31, .

Extreme Example

• 389965026819938 = 5569 (prime) + 389965026814369 (also prime)

Since 1742, others have tried to prove Goldbach’s Conjecture

• Euler noted that “1” is not considered to be prime and restated the conjecture back to Goldbach as “all positive even integers can be expressed as the sum of two primes”

• This alters the conjecture to read “every even number greater than or equal to four is the sum of two primes”

Proofs

• Goldbach’s conjecture has been researched and the majority of mathematicians believe the (strong) conjecture to be true.

• This is based on the statistical theory that the bigger the even number, the more likely is becomes that it can be written as the sum of two primes.

• In 1938, T. Estermann proved “that almost all even number are the sum of two primes.”

• Also in 1938, “N. Pipping verified (by hand) the conjecture for all even numbers less than or equal to 10,000.”

• In 1966, Chen Jing-run proved that “every sufficiently large even numbers can be written as the sum of two primes, or a prime and a semiprime.” (a number that is the product of two primes)

• In 1977, H. A. Pogorzelski circulated a proof, but his work has not been accepted.

• In March of 2000, T. Oliveira e Silva started distributing his computer search that has verified the conjecture up to 2 x 10 to the 17th power. His work is still ongoing.

Finally

• Even the works of mathematicians can use the dollar to drum up interest. In an effort to sell a book on Goldbach’s Conjecture, a British publisher offered a $1,000,000.00 prize for proof of the conjecture. The offer expired in 2000 and was unclaimed.