Arithmetic progressions, like the candles of Hanukkah or the gifts of ‘The Twelve Days of Christmas,’ can be powerful tools in number theory
ILLUSTRATION: TOMASZ WALENTA
By
Eugenia
Cheng
Dec.
5, 2018 10:28 a.m. ET
This week Jews around the world are lighting candles in
eight-branched menorahs (with one “attendant” candle to light the others) to
celebrate the festival of Hanukkah. As a symbol of the miracle of one night’s
worth of oil giving light for eight days, the candles are lighted in sequence
with one candle on the first night, two on the second and so on until the final
night.
In mathematics, this is called an arithmetic
progression—a sequence in which the numbers increase by the same amount at each
step. This contrasts with a geometric progression, in which the numbers are
successively multiplied by the same amount and grow exponentially. Arithmetic
progressions grow slowly and steadily, which might seem boring, but they have
far-reaching interest in pure mathematics.
Arithmetic progressions are involved in the study of
prime numbers, the elusive building blocks of number theory. A celebrated
theorem proved by Ben Green and Terence Tao in 2004 shows that the sequence of
prime numbers—numbers with exactly two factors, one and the number
itself—contains arithmetic progressions of unlimited length. The sequence 3, 5,
7, where the numbers increase by 2, has a length of 3, but it doesn’t go any
further since the next number in the progression would be 9, which isn’t prime.
To get a progression of prime numbers with a length of 10 we have to go up to
199, 409, 619, and so on, which has increments of 210.
The existence of longer and longer progressions is
surprising, because the prime numbers generally get more and more spaced out as
they get bigger. Exactly how the prime numbers are distributed is notoriously
difficult to pin down, though mathematicians have been trying for centuries.
A more straightforward process involving arithmetic
progressions is simply to add up the numbers. Soon we’ll hear everywhere “The
Twelve Days of Christmas,” the song involving more gifts every day until on the
12th day we have 12 drummers drumming, 11 pipers piping and so on, all the way
down to yet another partridge in a pear tree. How many total gifts do we end up
with?
The numbers are quite small, so we can add them up fairly
easily: 1+2+3+4, etc. But if the numbers were larger or the sequence longer, a
small trick would help. We could write out the numbers 1 to 12 in a row and
then write the same numbers in a row underneath them but backward, so that 12
will be under 1, 11 will be under 2 and so on.
Each vertical pair of numbers, it turns out, adds up to
the same thing: 13. There are 12 columns, so the total of all the numbers we’ve
written down must be 12x13=156. Because we wrote down our list of numbers
twice, the total we want is half of 156, that is, 78.
At one level this is just a neat trick for adding up a
list of numbers, but it also gives a taste of the field of combinatorics, which
deals with combinations of objects in different configurations, including
shapes, connected networks and codes. It often involves flipping the way we
think about something in order to do surprisingly efficient reasoning. Seeing
the trick of adding up arithmetic progressions is one of the first things that
attracted me to math when I was a child.
As for the application of arithmetic progressions to
prime numbers, life can go on if we don’t understand exactly how the prime
numbers are distributed. But the mathematician’s urge is to understand the
logical core of everything around us, especially the things that are basic on
the surface but mysterious deep down.
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