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
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
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
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.
MacArthur Fellow Yitang Zhang
John D. & Catherine T.
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.
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.
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.
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
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
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,
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
The Riemann Hypothesis
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.
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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