# Prove a Theorem, Win \$1,000,000!

— One generally doesn’t speak the words “prime numbers” and “seven-figure prizes” in the same breath.

But don’t tell that to the publishers of Uncle Petros and Goldbach’s Conjecture, an engaging first novel by Greek author Apostolos Doxiadis.

#### The Story Behind the Math

Before getting to the money, here’s a quick synopsis of the story: The narrator tells of his Uncle Petros, whom he initially thinks of as the eccentric black sheep of the family.

Slowly, Uncle Petros is revealed to be a character of complexity and nuance, having devoted his considerable mathematical talents and much of his life to a futile effort to prove a classic unsolved problem. His solitary efforts give one a taste of the delight and the despair of mathematical research.

Goldbach’s Conjecture, Uncle Petros’ holy grail, is startlingly simple to state:

Any even number greater than 2 is the sum of two prime numbers.

Remember that a prime number is a positive whole number that is divisible only by two numbers: itself and 1; thus 5 is a prime, but 6, which is divisible by 2 and 3, is not. The number 1 is not considered prime.

Check out the claim. Pick an even number at random and try to find two primes which add up to it. Certainly, 6 = 3 + 3, 20 = 13 + 7, and 97 + 23 =120. (This, of course, is not a proof.) The conjecture that this works for every even number greater than 2 was proposed in 1742 by Prussian mathematician Christian Goldbach. To this day it remains unproved despite the efforts of some of the world’s best mathematicians.

#### The Frustrations of Whole Numbers

 A Small Challenge For those with a desire to prove something, albeit something easier, try this. (Your reward will be the satisfaction of understanding, which is worth more than money.) Pick any 10 numbers between 1 and 100. There will always be two subsets of these 10 numbers whose sums are equal. Thus, for example, if you were to choose 51, 11, 81, 68, 73, 87, 23, 29, 25, 94, as I just did using a random number generator, you would soon observe that 25 + 51 + 29 = 94 + 11. The claim is that this works for every 10 numbers you choose. Prove it! Likewise, if you were to pick 20 whole numbers between 1 and 50,000, you would always find two subsets of these 20 numbers whose sums were equal.

Number theory, the branch of mathematics that studies prime numbers and other ethereal aspects of the integers (whole numbers), contains many problems that are easy to state and yet resistant, so far, to the efforts of all.

The Twin Primes Conjecture is another:

There are an infinite number of prime pairs, prime numbers that differ by 2.

Examples are 5 and 7, 11 and 13, 17 and 19, 29 and 31, and, presumably, infinitely many more.

Of more contemporary origin is the so-called Collatz Conjecture, sometimes called the 3x + 1 problem.

Choose any whole number. (Take 13, for example) If it is odd, multiply it by 3 and add 1. (3 times 13 plus 1 equals 40.) If it is even, divide it by 2. (40 divided by 2 is 20.) Continue this procedure with each resulting number and the conjecture is that the sequence thus generated always ends up 4, 2, 1, 4, 2, 1, 4, ... The sequence starting with 13, produces 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ....

Every number that’s been tried (up to about 27 quadrillion) ultimately cycles back to 4, 2, 1, but there is still no proof that every number does.

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