Apostolos Doxiadis - Uncle Petros and Goldbach

EVERY EVEN NUMBER GREATER THAN 2

IS THE SUM OF TWO PRIMES

At the time this book was completed, Goldbach's Conjecture was two hundred and fifty years old. To this day it remains unproven.

I wish to thank Professors Ken Ribet and Keith Conrad, who carefully read the revised manuscript and corrected numerous mistakes, as well as Dr Kevin Buzzard for the clarification of various points – obviously, any remaining mathematical flaws are my own. Also my sister, Cali Doxiadis, for her invaluable linguistic and editorial advice.

APOSTOLOS DOXIADIS

[1]A method for locating the primes, invented by the Greek mathematician Eratosthenes

[2]According to the American system, a student can go through the first two years of university without being obliged to declare an area of major concentration for his degree or, if he does so, is free to change his mind until the beginning of the Junior (third) year.

[3]In fact, Christian Goldbach's letter of 1742 contains the conjecture that 'every integer can be expressed as the sum of three primes'. However, as (if this is true) one of the three such primes expressing even numbers will be 2 (the addition of three odd primes would be of necessity odd, and 2 is the only even prime number), it is an obvious corollary that every even number is the sum of two primes. Ironically, it was not Goldbach but Euler who phrased the conjecture that bears the other's name – a little known fact, even among mathematicians.

[4]The main purpose of this narrative is not autobiographical, so I will not burden the reader further with details of my own mathematical progress. (To satisfy the curious I could sum it up as 'slow but steady'.) Henceforth, my own story will be referred to only to the extent to which it is relevant to that of Uncle Petros.

[5]Principia Mathematica: the monumental work of logicians Russell and Whitehead, first published in 1910, in which they attempt the titanic task of founding the edifice of mathematical theories on the firm foundations of logic.

[6]The largest such pair known today is almost inconceivably enormous: 835335^39014 +/-1.

[7]Let k be a given integer. The set (k + 2)! + 2, (k + 2)! + 3, (k + 2)! + 4… (k + 2)! + (k + 1), (k + 2)! + (k + 2) contains k integers none of which is prime, since each is divisible by 2,3,4…, k + 1, k + 2 respectively. (The symbol k!, also known as 'k factorial', means the product of all the integers from 1 to k.)

[8]Numbers of the form a + bi, where a, b are real numbers and i is the 'imaginary' square root of -1.

[9]This states that any odd number greater than 5 is the sum of three primes.

[10]In his seminal work The Nature of Mathematical Discovery, Henri Poincare demolishes the myth of the mathematician as a totally rational being. With examples drawn from history, as well as from his own research experience, he places special emphasis on the role of the unconscious in research. Often, he says, great discoveries happen unexpectedly, in a flash of revelation that comes in a moment of repose – of course, these can occur only to minds that are otherwise prepared through endless months or years of conscious work. It is in this aspect of the workings of a mathematician's mind that revelatory dreams can play an important role, sometimes providing the route through which the unconscious announces its conclusions to the conscious mind.

[11]It was Fermat who first stated the general form, obviously generalizing from age-old observations that this was true of the first four values of n, i.e. 2^2^1 +1 = 5, 2^2^2 +1 = 17, 2^2^3 +1 = 257, 2^2^4 + 1 = 65537, all prime. However, it was later shown that for n = 5, 2^2^5 +1 equals 4294967297, a number which is composite, since it's divisible by the primes 641 and 6700417. Conjectures are not always proved correct!

[12]Indeed, 1729 = 12^3 + 1^3 = 10^3 + 9^3, a property which does not apply for any smaller integer.

[13]C.Cavafy,'Ithaca'

[14]The great unsolved problems stated by David Hubert at the International Congress of Mathematicians in 1900. Some, like the Eighth Problem (the Riemann Hypothesis) are still outstanding, but in others there has been progress and a few have been completely solved – as, for example, the Fifth, proved by Gleason, Montgomery and Zippen; the Tenth, by Davis, Robinson and Matijasevic; the Fourteenth, proved false by Nagata; the Twenty-second, solved by Deligne

[15]Gödel subsequently ended his own life, in 1978, while being treated for urinary tract problems at the Princeton County Hospital. His method of suicide was, like his great theorem, highly original: he died of malnutrition, having refused all food for over a month, convinced that his doctors were trying to poison him

[16]Mystery-solutions to famous problems by charlatans are two-a-penny

[17]Fermat's Last Theorem was, amazingly, proved in 1993. Gerhard Frey first proposed that the problem could possibly be reduced to an unproven hypothesis in the theory of elliptic curves, called the Taniyama-Shimura Conjecture, an insight later conclusively proven by Ken Ribet. The crucial proof of the Taniyama-Shimura Conjecture itself (and thus, as its corollary, Fermat's Last Theorem) was achieved by Andrew Wiles; in the final stage of his work he collaborated with Richard Taylor