Prime numbers: the 271 year old puzzle resolved

STORY BY Artem Kaznatcheev

Published: May 13, 2013

The odd Goldbach conjecture, a two-hundred and seventy-one year open problem of mathematics, has been resolved. Earlier today, H.A. Helfgott proved that any odd number greater than 5 can be written as the sum of 3 primes.

Number theory is the Queen of Mathematics, and the darling of number theory is the study of primes. A prime is a positive number that can only be divided by 1 and itself. A deceptively simple definition, but one that has captured the imagination of mathematicians since Euclid published his Elements around 300 BC. Through the years, thinkers have retreated to the sanctuary of prime numbers to stretch their minds and exercise their intellect in what was considered the most ivory tower of pursuits. The field was believed to be innocent of application, and free of the taint of industry and war. In 1940, struggling with the turmoil around him, the famous number-theorist G.H. Hardy wrote the historic words:

"I have never done anything 'useful'. No discovery of mine has made, or is likely to make,   directly or indirectly, for good or ill, the least difference to the amenity of the world."

He was writing about his study of the primes. Little could he foresee that only 30 years later, the darling of mathematics would go on to form the key ingredient of the RSA algorithm for public key cryptography. Today, the difficulty of separating a number into the primes that multiply to form it secures all our private transactions on the internet.

Goldbach

Of course, multiplication is not the only way to split a number, our other choice is to use addition. In 1742, Prussian mathematician Christian Goldbach in a the margins of a letter of Leonhard Euler doodled a conjecture: every integer greater than 2 can be written as the sum of three primes. In the quarter-millennium since, the conjecture has been split into two parts, the strong and weak. The strong Goldbach conjecture states that every even number greater than 4 can be written as the sum of two primes, and the weak Goldbach conjecture states that every odd number greater than 5 can be written as the sum of three primes. The best mathematicians in the world have obsessed with both conjectures for centuries.

On Monday, May 13th, Harald Andrés Helfgott of the École Normale Supérieure – Paris posted a 133 page mansucript proving the weak Goldbach conjecture. His argument applies the circle method first developed by Hardy in 1917 (working with J.E. Littlewood  and independently by I. M. Vinogradov). Mathematicians have been itching to close this chapter, and have inched towards a proof for years. Last year the result was close to resolve with Fields medalist (the mathematics equivalent of a Nobel Prize) Terence Tao showing that any odd integer is the sum of at most 5 primes. Now the conjecture is solved, and although it still has to go through the formalities of publication, Helfgott's preprint is endorsed and believed to be true by top mathematicians, Tao among them.

 

Co-written by Artem Kaznatcheev and Kate Zen.

Kate Zen is a freelance writer from the Boogie Down Bronx, currently living in Montreal, Canada.

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