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Home >Opinion >Columns >The charm of an endless series of series

In parallel with various other subjects I’ve written about in this column, I’ve often wanted to write about series. “Series", meaning sequences of numbers. The great charm of numbers is the endless ways you can extract sequences from them, the logic that applies, and the search for meaning in that series. (I’ll use “sequence" and “series" interchangeably).

This is no exotic notion. It’s embodied in a hundred “what’s-the-next-number-in-this-sequence?" puzzles you’ve solved over the years. Like, what’s the next number in each of these?

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* 1, 2, 3, 4, 5 ...

* 1, 2, 4, 8, 16 ...

* 0, 1, 1, 2, 3, 5 ...

* 1, 4, 19, 6, 108, 37 ...

You quickly answered the first two: 6 and 32, respectively. The third is the famous Hemachandra or Fibonacci series, in which each number is the sum of the previous two. The fourth? You’re welcome to search, and you may even find some logic. But honestly, I just threw those numbers out at random.

Here are a few intriguing sequences, dealing only in our “natural numbers", the positive integers.

#1: 1, 2, 3, 4, 5 ... (again).

What’s interesting here? These are just the regular numbers we use daily. Yet that’s the point: natural numbers are the foundation of our entire number system, and that’s why they are interesting. They go on endlessly, they are regularly spaced, they allow the extractions I mentioned above, and much more.

#2: 0, 1, 1, 2, 3, 5 ... (again).

Such a simple rule that defines this sequence: start with 0 and 1, add the two most recent entries to get the next. And yet there are all kinds of gems to discover here. The ratio between two successive numbers gets closer and closer to the number j (phi), usually called the “golden ratio", 1.618... Closer and closer, but never quite there, because jis an irrational number, meaning it cannot be expressed as the ratio of two integers. And get this: start with any two integers and apply the rule, and you’ll find j popping up in the same way. You don’t need to start with 0 and 1.

More nuggets? You will find the series appears in sunflower heads (really). If you count the ways a bee— a mathematically-inclined bee, ok — might make its mathematical way from one cell to another in a honeycomb, this sequence appears again. Every 3rd number in the series is a multiple of 3; every 4th, a multiple of 4; every 5th, a multiple of 5; and so on.

You can also use it in party tricks that never fail to amaze. One goes like this: ask your audience for two numbers at random, and apply the rule to generate the series. At the seventh number, multiply it by 11 and announce it as a prediction. Stop at the tenth and sum them all. Voilà! The total will match your prediction.

Still, Hemchandra/Fibonacci is famous enough. Onward— to some more obscure series.

#3: 1, 2, 4, 7, 11, 16 ...

The name is the charm. You can quickly divine the logic governing the series. But it’s called the Lazy Caterer’s sequence, and there’s a reason. These are numbers generated by cutting a food item—a pizza or a pancake, something round and flat— to produce as many pieces as possible, not necessarily equal.

If we start the numbering from 0, it goes like this. Apply zero cuts to the pizza, and you have just 1 piece (the whole pizza, didn’t you know). One straight-line cut produces 2 pieces. Two, a maximum of 4 pieces, if you slice across the previous cut. For every subsequent cut, the way to maximize the count of pieces is to slice across all previous cuts. You must make sure not to slice through an intersection of two cuts, because that loses one potential piece. Slice like this each time, and each slice adds as many pieces as the serial number of the slice. Third cut, three new pieces, total of 7. Fourth cut, four new pieces, total of 11. Etc.

#4: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127 ...

Impossible to work out the logic here? But a clue: If you know the first few powers of 2, you might recognize the first few numbers here as being one less than powers of 2 (4, 8, 32, 128, 8192). That’s good to note, because that’s in fact just what these numbers are: one less than particular powers of 2. But which particular powers?

These: 2, 3, 5, 7, 13, 17, 19 ... aha! A sequence by itself, and wait, all those are prime numbers! Yet 11 is missing. And in fact, the next number here is 31, which means the primes 23 and 29 are missing. So which particular prime numbers?

This is how sequence #4 is generated: Raise 2 to a prime power and subtract 1. If the answer is itself prime, it’s part of the sequence. These are the so-called Mersenne primes, and while mathematicians hypothesize that they are infinite in number, we currently know only 51 of them. The 51st is formed by raising 2 to the 82,589,933rd power and subtracting 1. What we get is a gargantuan beast with 24,862,048 digits. To put that in perspective: if I tried to type it out, its digits would fill about 250 columns with just digits. I could submit those and take a break from column writing for the next 5 years, though admittedly this esteemed publication may choose not to pay.

That 51st entry in this sequence is currently the largest-known prime number. In fact, this series of Mersennes has given us the eight largest-known primes; and the last 17 times a record has been set for the largest-known prime, it has been a Mersenne each time. All that is because this method— take a prime number, raise 2 to that power and subtract 1— at least gives us a number whose primality is easier to check than others.

#5: 1, 2, 4, 8, 13, 21, 31 ...

It’s possibly even more difficult to find logic here just by looking at the sequence. Start with 1. Any given entry after that, then, is the smallest integer greater than the previous such that if you made pairs of the entries up to and including that one and summed the pairs, you would not have any repeating totals.

Got that?

The second entry is 2, because then we’d have 1 + 1 = 2, 1 + 2 = 3 and 2 + 2 = 4: 3 different sums. The third entry cannot be 3, because we would have 2 + 2 = 1 + 3 = 4. Try 4 for the third entry, and we have 1 + 1 = 2, 1 + 2 = 3, 2 + 2 = 4, 1 + 4 = 5, 2 + 4 = 6 and 4 + 4 = 8: 6 different sums, so 4 it is. And on we go.

This curiosity is called the Mian-Chowla sequence, for Abdul Majid Mian, a physicist at the University of Peshawar, and Sarvadaman Chowla, a mathematician born and raised in Britain. They published a paper about it together in the Proceedings of the National Academy of Sciences, India, in 1944.

There you have it: five number sequences of varying interest and opacity. What about utility? Well, the natural numbers have occasional uses, sure. And very large primes are critical in cryptography. But the real charm in these, for me certainly, is what they say about mathematicians: how they constantly play with numbers, look for patterns. Yet you can too. Dream up your own extraction and you might soon have a series named for you.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun