q=p1×p2×p3×…×pn+1

Robert Neubecker for Quanta Magazine

Yours truly loves physics and math from the lay perspective. Being math challenged, I am constantly amazed as to how math is the short hand needed to describe reality. At the same time, the divine obsession of mathematicians to solve arcane problems seemingly disconnected from quotidian activities of society endlessly fascinates as seen by a recent piece in Quanta Magazine, a true garden of earthly delights from this writer's perspective. :)

To whit ...

If you’ve been following the math news this month, you know that the 35-year-old number theorist James Maynard won a Fields Medal — the highest honor for a mathematician. Maynard likes math questions that “are simple enough to explain to a high school student but hard enough to stump mathematicians for centuries,” Quanta reported, and one of those simple questions is this: As you move out along the number line, must there always be prime numbers that are close together?

You may have noticed that mathematicians are obsessed with prime numbers. What draws them in? Maybe it’s the fact that prime numbers embody some of math’s most fundamental structures and mysteries. The primes map out the universe of multiplication by allowing us to classify and categorize every number with a unique factorization. But even though humans have been playing with primes since the dawn of multiplication, we still aren’t exactly sure where primes will pop up, how spread out they are, or how close they must be. As far as we know, prime numbers follow no simple pattern.

Something's afoot. :)

To appreciate this question, let’s start with one of the first intriguing facts an aspiring number enthusiast learns: There are infinitely many prime numbers. Euclid proved this 2,000 years ago using one of the most famous proofs by contradiction in all of math history. He started by assuming that there are only finitely many primes and imagined all n of them in a list:

p1, p2, p3, …, pn.

Then he did something clever: He thought about the number 

q=p1×p2×p3×…×pn+1.

Notice that q can’t be on the list of primes, because it’s bigger than everything on the list. So if a finite list of primes exists, this number q can’t be prime. But if q is not a prime, it must be divisible by something other than itself and 1. This, in turn, means that q must be divisible by some prime on the list, but because of the way q is constructed, dividing q by anything on the list leaves a remainder of 1. So apparently q is neither prime nor divisible by any prime, which is a contradiction that results from assuming that there are only finitely many primes. Therefore, to avoid this contradiction, there must in fact be infinitely many primes.

Sorta like This sentence is false or ...

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