Goldbach Biography
Goldbach's full name was Christian Goldbach. He was born March 18, 1690 in Königsberg, Brandenburg-Prussia, a region near present-day Poland. His father was a pastor. When he became old enough, he began studying at the Royal Albertus University. Also known as the University of Königsberg, it was founded in 1944 by Duke Albert of Prussia. After his studies were finished, from 1710 to 1724 he travelled around Europe, visiting Germany, Italy, Holland, France, and England. During these trips and his brief visits home, he acquainted himself with Georg Bernhard Bilfinger, Nicholas I. Bernoulli, Jakob Hermann, Gottfried Leibniz, Leonhard Euler, and several other prominent mathematicians. When he finally settled in St. Petersburg in 1725, he taught mathematics and its history at the St. Petersburg Academy of Sciences. Three years later, in 1728, he became the tutor for the current Tsar of Russia, Peter II. In 1742, he got a job at the Russian Ministry of Foreign Affairs, where he worked until his retirement.
Goldbach is recognized today mostly for ideas he expressed in his conservations with Euler, Bernoulli, and Leibniz. He extensively studied perfect powers, which are positive integers that are able to be expressed as a power of another positive integer. An example of a perfect power is 27, which can be expressed as 3^3. One of Goldbach's theorems on perfect powers was one he had worked on with Euler, which stated that the sum of 1/(p - 1) equals 1 over the set of perfect powers, which all eventually equals 1:
This theorem is known as the Goldbach-Euler theorem. It first appeared in 1737 in a paper written by Euler called "Variæ observationes circa series infinitas". Euler and Goldbach formulated the theorem through a series of letters to each other, with Goldbach’s final letter to Euler finally producing the end result. Another subject Christian Goldbach studied was Fermat numbers, named after Pierre de Fermat, the first mathematician to study them. A Fermat number is a positive integer of this form: N in this form is a nonnegative integer, meaning it is greater than or equal to zero. The first several Fermat numbers are 3, 5, 17, 257, 65537, 4294967297, and 18446744073709551617. In addition, if a number is a prime of the form 2n + 1, it is a Fermat prime. The only Fermat primes that mathematicians know of are F0, F1, F2, F3, and F4. Goldbach’s theorem of Fermat numbers said that no two Fermat numbers share a common factor that is larger than 1.
However, his most famous work is contained in a letter of his written in 1742 to Leonhard Euler, in which he proposed a conjecture, now called Goldbach's conjecture. Today, Goldbach’s conjecture is still one of the most famous unsolved problems in math. In his conjecture, Goldbach claimed that every even integer greater than 2 is the sum of 2 prime numbers. The expression of this is n = p + q, n being every even integer greater than 2, p and q being prime numbers. Another way of stating this is saying that every even integer greater than 2 is a Goldbach number, since a Goldbach number is one of these numbers that is proven to be the sum of 2 prime numbers. So far, it has been proven by modern mathematicians that every integer greater than 2 up to 4 × 1018 is a Goldbach number, or the sum of 2 prime numbers. They proved this by creating a Goldbach partition for each number, an expression of the theory. Some examples of Goldbach partitions are: 2+2=4, 3+3=6, 3+5=8, and 5+5 or 3+7=10. In his letter to Euler, Goldbach wrote notes about his thoughts on his conjecture in the body of the letter and also on the margins. He first said: “Every integer which 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,” (Wikipedia 1). Units are single parts that constitute a whole. Later in the margins of his letter, he said: “every integer greater than 2 can be written as the sum of three primes,” (Wikipedia 1). This second sentence is the same as the present-day Goldbach conjecture. Back then, mathematicians considered 1 to be a prime number, since it can only be factored by itself and one. The number 1 was included in a few Goldbach partition, making Goldbach numbers the sum of three prime numbers. Now, we no longer consider 1 a prime number, as there are too many theorems that use prime numbers but exclude 1. Today, we consider the number 1 a unit, and the only positive integer that is divisible by one positive integer, while prime numbers are divisible by 2, composite numbers are divisible by more than 2, and 0 is divisible by all positive integers. Instead of specifying the exception of 1 in all of these theorems, it is easier to say that 1 is not prime. The modern version of Goldbach’s statement in the margins, excluding 1, is every integer that is greater than 5 is the sum of 2 primes. For example, the number 14 is the sum of 2 primes, 7 and 7, or the number 6 is the sum of 2 primes, 3 and 3. In following letters, Goldbach reverted to his original conjecture, restating that every even integer greater than 2 is the sum of 2 primes. Euler agreed, and even said “I regard (this) as a completely certain theorem, although I cannot prove it,” (Caldwell). This conjecture is known today as the ‘strong’ Goldbach conjecture. It implies another conjecture, the ‘weak’ Goldbach conjecture, also known as the ternary Goldbach problem, odd Goldbach conjecture, or the 3-primes problem. This ‘weak’ conjecture states that every odd integer that is greater than 5 can be written as the sum of three primes, allowing that a prime may be used more than once in the same sum. This conjecture is called weak because its validity is dependent on the validity of the original Goldbach conjecture, the ‘strong’ one. Even still, in 2013 a Peruvian mathematician named Harald Helfgott was able to prove Goldbach’s ‘weak’ conjecture. Him and other mathematicians like him continue to attempt to prove Goldbach’s original conjecture, the ‘strong’ one. Goldbach lived until his death at the age of 74. He died on November 20th, 1764, in Moscow, Russia.
Sources:
http://en.m.wikipedia.org/wiki/Christian_Goldbach
http://sweet.ua.pt/tos/goldbach.html#r2
http://mathforum.org/library/drmath/view/57058.html
http://en.wikipedia.org/wiki/University_of_Königsberg
http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture
http://en.wikipedia.org/wiki/Goldbach–Euler_theorem
http://en.wikipedia.org/wiki/Fermat_number#Basic_properties
http://en.wikipedia.org/wiki/Fermat_number
Bibliography:
Wikipedia. "Goldbach's Conjecture." Wikipedia. Wikimedia Foundation, n.d.
Web. 10 Nov. 2014.
Caldwell, Chris (2008). "Goldbach's conjecture". Retrieved 2008-08-13.
Goldbach is recognized today mostly for ideas he expressed in his conservations with Euler, Bernoulli, and Leibniz. He extensively studied perfect powers, which are positive integers that are able to be expressed as a power of another positive integer. An example of a perfect power is 27, which can be expressed as 3^3. One of Goldbach's theorems on perfect powers was one he had worked on with Euler, which stated that the sum of 1/(p - 1) equals 1 over the set of perfect powers, which all eventually equals 1:
This theorem is known as the Goldbach-Euler theorem. It first appeared in 1737 in a paper written by Euler called "Variæ observationes circa series infinitas". Euler and Goldbach formulated the theorem through a series of letters to each other, with Goldbach’s final letter to Euler finally producing the end result. Another subject Christian Goldbach studied was Fermat numbers, named after Pierre de Fermat, the first mathematician to study them. A Fermat number is a positive integer of this form: N in this form is a nonnegative integer, meaning it is greater than or equal to zero. The first several Fermat numbers are 3, 5, 17, 257, 65537, 4294967297, and 18446744073709551617. In addition, if a number is a prime of the form 2n + 1, it is a Fermat prime. The only Fermat primes that mathematicians know of are F0, F1, F2, F3, and F4. Goldbach’s theorem of Fermat numbers said that no two Fermat numbers share a common factor that is larger than 1.
However, his most famous work is contained in a letter of his written in 1742 to Leonhard Euler, in which he proposed a conjecture, now called Goldbach's conjecture. Today, Goldbach’s conjecture is still one of the most famous unsolved problems in math. In his conjecture, Goldbach claimed that every even integer greater than 2 is the sum of 2 prime numbers. The expression of this is n = p + q, n being every even integer greater than 2, p and q being prime numbers. Another way of stating this is saying that every even integer greater than 2 is a Goldbach number, since a Goldbach number is one of these numbers that is proven to be the sum of 2 prime numbers. So far, it has been proven by modern mathematicians that every integer greater than 2 up to 4 × 1018 is a Goldbach number, or the sum of 2 prime numbers. They proved this by creating a Goldbach partition for each number, an expression of the theory. Some examples of Goldbach partitions are: 2+2=4, 3+3=6, 3+5=8, and 5+5 or 3+7=10. In his letter to Euler, Goldbach wrote notes about his thoughts on his conjecture in the body of the letter and also on the margins. He first said: “Every integer which 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,” (Wikipedia 1). Units are single parts that constitute a whole. Later in the margins of his letter, he said: “every integer greater than 2 can be written as the sum of three primes,” (Wikipedia 1). This second sentence is the same as the present-day Goldbach conjecture. Back then, mathematicians considered 1 to be a prime number, since it can only be factored by itself and one. The number 1 was included in a few Goldbach partition, making Goldbach numbers the sum of three prime numbers. Now, we no longer consider 1 a prime number, as there are too many theorems that use prime numbers but exclude 1. Today, we consider the number 1 a unit, and the only positive integer that is divisible by one positive integer, while prime numbers are divisible by 2, composite numbers are divisible by more than 2, and 0 is divisible by all positive integers. Instead of specifying the exception of 1 in all of these theorems, it is easier to say that 1 is not prime. The modern version of Goldbach’s statement in the margins, excluding 1, is every integer that is greater than 5 is the sum of 2 primes. For example, the number 14 is the sum of 2 primes, 7 and 7, or the number 6 is the sum of 2 primes, 3 and 3. In following letters, Goldbach reverted to his original conjecture, restating that every even integer greater than 2 is the sum of 2 primes. Euler agreed, and even said “I regard (this) as a completely certain theorem, although I cannot prove it,” (Caldwell). This conjecture is known today as the ‘strong’ Goldbach conjecture. It implies another conjecture, the ‘weak’ Goldbach conjecture, also known as the ternary Goldbach problem, odd Goldbach conjecture, or the 3-primes problem. This ‘weak’ conjecture states that every odd integer that is greater than 5 can be written as the sum of three primes, allowing that a prime may be used more than once in the same sum. This conjecture is called weak because its validity is dependent on the validity of the original Goldbach conjecture, the ‘strong’ one. Even still, in 2013 a Peruvian mathematician named Harald Helfgott was able to prove Goldbach’s ‘weak’ conjecture. Him and other mathematicians like him continue to attempt to prove Goldbach’s original conjecture, the ‘strong’ one. Goldbach lived until his death at the age of 74. He died on November 20th, 1764, in Moscow, Russia.
Sources:
http://en.m.wikipedia.org/wiki/Christian_Goldbach
http://sweet.ua.pt/tos/goldbach.html#r2
http://mathforum.org/library/drmath/view/57058.html
http://en.wikipedia.org/wiki/University_of_Königsberg
http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture
http://en.wikipedia.org/wiki/Goldbach–Euler_theorem
http://en.wikipedia.org/wiki/Fermat_number#Basic_properties
http://en.wikipedia.org/wiki/Fermat_number
Bibliography:
Wikipedia. "Goldbach's Conjecture." Wikipedia. Wikimedia Foundation, n.d.
Web. 10 Nov. 2014.
Caldwell, Chris (2008). "Goldbach's conjecture". Retrieved 2008-08-13.