Euclid

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Prime numbers are one of the most basic topics of study in the

branch of mathematics called number theory.

Primes are numbers that can only be evenly divided by themselves

and 1. For example, 7 is a prime number since I’m left with a

remainder or a fractional component if I divide 7 by anything

other than itself or 1. 6 is not a prime because I can divide 6

by 2 and get 3.

One of the reasons primes are important in number theory is that

they are, in a certain sense, the building blocks of the natural

numbers. The

fundamental theorem of arithmetic (the name of which

indicates its basic importance) states that any number can be

factored into a unique list of primes. 12 = 2 x 2 x 3, 50 = 5 x 5

x 2, 69 = 3 x 23.

Studying numbers, then, basically amounts to studying the

properties of prime numbers. Mathematicians have, over the

millennia, figured out quite a bit about the prime numbers. One

of Euclid’s most famous proofs shows that there are infinitely

many primes.

The basic

idea of the proof is that if there were only finitely many

primes, and we had a list of all of those prime numbers, we could

multiply them all together and add 1, creating a new number that

isn’t divisible by any of the prime numbers on our list. That

number would either itself be a prime number not on our list, or

would have a prime divisor not on our list. Either way, we

contradict the idea that there could be a finite list of primes,

and so there have to be infinitely many.

In the nineteenth century, mathematicians proved the Prime

Number Theorem. Given some large natural number, the theorem

gives a rough estimate for how many numbers smaller than the

given number are prime. Primes get rarer among larger numbers

according to a particular approximate formula.

Despite all the things we know about prime numbers, there are

plenty of deceptively simple conjectures about primes that have

not yet been either proven or disproven. Here are some of those

conjectures.

##
Twin Primes

Twin primes

are pairs of prime numbers that have just one number between

them: 5 and 7, 11 and 13, and 29 and 31. The twin primes

conjecture is that there are infinitely many pairs of twin primes

among the infinitely many prime numbers.

Most mathematicians think that the conjecture should be true:

while prime numbers get rarer as numbers get larger, number

theorists’ experience and intuition with primes suggests that

twin prime pairs should still pop up from time to time. Despite

this, the conjecture has not yet been proven or disproven.

Mathematician and

MacArthur Fellow Yitang Zhang

John D. & Catherine T.

MacArthur Foundation

After remaining a completely open question for centuries, in

spring 2013, University of New Hampshire mathematician Yitang

Zhang made a breakthrough in the problem,

for which he was awarded a MacArthur "Genius" fellowship in

September 2014. While still not proving the twin primes

conjecture itself, Zhang invented a novel technique that showed

that there are infinitely many pairs of prime numbers with no

more than 70,000,000 numbers between them. That’s a huge number,

but it was the first such finite limit on distances between

primes that had ever been discovered.

In fall 2013, a large group of mathematicians built on Zhang’s

work and similar results and collaboratively found smaller and

smaller bounds, eventually proving that there are infinitely many

pairs of primes with at most 246 numbers between them.

##
Goldbach’s Conjecture

This is another simply stated problem. Goldbach’s

Conjecture says that every even number larger than two can be

written as the sum of two prime numbers. This certainly holds

true for smaller numbers: 4 = 2 + 2, 8 = 5 + 3, 20 = 13 + 7, but

it hasn’t been proven for all even numbers.

Researchers armed

with 21st century computers and well-designed programs have

verified the conjecture for even numbers up to

4,000,000,000,000,000,000. This is pretty good evidence for the

conjecture, but in mathematics, saying that a conjecture holds

for all numbers smaller than some ludicrously high finite bound

is not enough to say that it holds for all numbers.

##
Palindromic Primes

Palindromes in English are words or sentences that read the same

forwards and backwards. The word "radar" and the phrase "A man, a

plan, a canal: Panama" are both palindromes.

Similarly, palindromic prime numbers are primes whose decimal

expansions read the same forwards or backwards. 11, 101, and

16561 are some examples of

palindromic primes.

My personal favorite prime number is

Belphegor’s Prime: 1000000000000066600000000000001. That’s a

1, followed by 13 0’s, followed by a 666, followed by 13 more

0’s, followed by a closing 1. Given that 666 is the "number of

the beast" according to

Revelation 13:18, and 13 has a slew of superstitions attached

to it, this is probably the unluckiest prime number possible in

the decimal number system.

"The Number of the Beast

is 666" – William Blake

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As with the twin primes, it’s currently unknown if there are

infinitely many palindromic primes. The palindromic primes are a

less active area of mathematical research than the twin primes,

however.

Problems like Goldbach’s Conjecture and the twin primes

conjecture rely solely on the structure and distribution of the

primes themselves. But palindromic primes depend on the particular

number system being used: binary palindromes are completely

different than decimal palindromes. The prime number written in

decimal as 31 is written in binary as 11111. In binary, this

prime is a palindrome; in decimal, it is not.

While mathematicians do study

the palindromic primes, and have come up with results like that

primes are rare

among palindromes regardless of the number system being used,

more effort in number theory is dedicated to problems that focus

mostly on the properties of primes independent of their

representations.

##
The Riemann Hypothesis

The Riemann

Hypothesis is one of the Millennium Prize Problems, a set of

the most important open problems in mathematics. Solving one of

these problems brings with it a prize of $1,000,000.

The Riemann Hypothesis involves

an extension to the Prime Number Theorem mentioned above. That

theorem gives a formula for the approximate number of primes

smaller than some given large number. The Riemann Hypothesis

gives a more specific result, providing a formula showing how

accurate that estimate will be.

Bernhard

Riemann

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The great nineteenth century mathematician Bernhard Riemann

connected that accuracy bound to a special function on the

complex number plane. The actual Riemann Hypothesis states that

all of the points on the complex plane where that function equals

zero fall along a particular line in the plane. Should that be

the case, the accuracy bound would also be true.

As with the other problems on

this list, there is a good amount of numerical evidence for the

Riemann Hypothesis, and most mathematicians believe it to be

true. Mathematicians have tested billions of the zero points of

the function and found all of them to fall on that line.

Also like the other problems

we’ve looked at, there is not yet a full blown proof of the

hypothesis. In each of these cases, while most mathematicians

believe these conjectures to be true, and there is a good bit of

empirical evidence for the conjectures, the search for a full

blown proof continues.

This seemingly obsessive

behavior on the part of mathematicians is partially because

rigorous proof is one of the main goals of mathematics, but also

because any proof of the twin primes conjecture, or of the

Riemann Hypothesis, would likely involve radically new

mathematical techniques and insights, potentially leading to

entirely new avenues of research and ideas to explore. In

mathematics, it’s often the case that the journey to finding a

proof is at least as interesting as the result itself.

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