Goldbach's Conjecture

 

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Christian Goldbach was born in March 1690 in Königsberg, Prussia (now Kaliningrad, Russia), and died in 1764 in Moscow, Russia. When he was 35 Goldbach became a professor of mathematics and a historian at St. Petersburg. He went to Moscow in 1728 to be a tutor to Tsar Peter II. Goldbach knew many mathematicians around Europe.

In 1742 he wrote to Euler conjecturing that every even integer greater than 2 can be represented as a sum of 2 primes.

n = p1 + p2

This conjecture has not yet been proved or disproved.

This conjecture is equivalent to saying that every integer greater than 5 is the sum of three primes.

Copy of Goldbach's letter to Euler in which he conjectures, dated 7 th July 1742.

 

Euler responded to Goldbach saying that

"There is little doubt that this result is true... that every even number is a sum of two primes, I consider [this] an entirely certain theorem in spite of that I am not able to demonstrate it."

Ivan Matveevich Vinogradov was another Russian mathematician (1891-1983) who showed that if we look at 'sufficiently' large odd integers, we deduce that they can be written as the sum of at most three primes. From this follows that every sufficiently large integer (not necessarily odd) is the sum of at most four primes. One result of Vinogradov's work is that we take that Goldbach's Conjecture holds true for almost all even integers.

Try for yourself:

1. get any even number

8=5+3

20=13+7

while odd number can be expressed as sum of three primes.

21=11+7+3 (see Vinogradov's addition above).

 

   

See some other unsolved problems from the history of mathematics, or some solved ones!

Have a look at other pages on primes here.

Main number page

Prime numbers

Largest Prime Number known

See Eratosthenes' Prime Number sieve and download some worksheets: 100, 200 or 500 number sieve.

What people thought of primes through the history

Prime number (as the one defined by Aristotle, Euclid and Theon of Smyrna) is a number "measured by no number but by an unit alone" Iambilicus said that a prime number is also called "odd times odd".

Prime number was apparently first described by Pythagoras.

Iamblichus writes that Thymaridas called a prime number rectilinear since it can only be represented one-dimensionally.

In English language 'prime number' is first to be found in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

Some older textbooks include 1 as a prime number.

In his Algebra (1770), Euler did not consider 1 a prime [William C. Waterhouse].

 

 

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