Complements #4 : Prime Numbers

silly_prime_joke

Prime numbers get a really rough deal in schools. They’re often brushed over or bunched together with triangle and square numbers for a single hour lesson if they’re lucky.

“A prime number is divisible by itself and one, such as? … Great, that’s that one sorted”

Sigh.

Prime numbers deserve more attention. Below is a nice version (ish) of the Fundamental Theorem of Arithmetic, proven in 300BC by Euclid the Rock God of Maths (unofficial title). Not only that, but it’s also a kind of version of the Sieve of Eratosthenes. The theorem states that every whole number greater than 1 is either a prime or a product of primes. In other words each natural number is built from a unique set of prime numbers (You can of course, derive a number’s unique combination of prime factors by using a factor tree blah blah you knew that bit.). Whereas the Sieve of Eratosthenes is a (fairly laborious) method of finding primes, by finding all the multiples of a prime (known as composite numbers), then moving along to the next number that wasn’t marked by any prime before it. It was quite handy back in the day to find low prime numbers.

pr

Sieve_of_Eratosthenes_animation

There are other sieves too. The sieve of Atkin who later went on to create a crazy diet (err, wrong Atkin), the sieve of Sundaram and Euler’s sieve, which is basically the same as the Sieve of Eratosthenes. All of these sieves suffer the same stumbling setback – they take ages to find primes much larger than a few hundred. And primes become much more sporadic the further up the number line you dare to tread.

Anyhoo, the fact that prime numbers are essentially the basis of all other numbers makes them rather special. They are considered the ‘atoms’ of mathematics. If we knew them all we could make a kind of Periodic Table for maths. Sadly we don’t. But that’s also why primes are so fun.

Their mystery also makes them appealing. Is there a prime pattern?

If you fill out a 10 x 10 grid with prime numbers, you’d be forgiven for thinking there’s nothing spectacular to behold:

aoKCC

(meh)

but if you change that grid to 6 across …

pat

Oooh… patterny. Everything (bar 2 primes) is now in two distinct columns. That’s because primes can’t be multiples of 3, or even numbers, so that leaves numbers one above and one below multiples of 6. Of course not ALL numbers one above or one below 6 are primes otherwise we’d have this whole prime thing pretty nailed by now.

But there IS a pattern. And it’s so close… yet so far.

Throughout history people have cracked pieces of the puzzle. One of the most famous examples is by Marin Mersenne in the 17th Century. He claimed to have found a large quantity of prime numbers using the formula 049087cbf961a9d755e7fa8c067c49bb although a lot of them were later proven to not be prime numbers. However, any number that fits that formula and *isn’t* prime, still gets the fancy title of ‘Mersenne Number’.  The ones that *are* prime numbers get the fancier title of ‘Mersenne Prime‘. Despite the minor errors along the way, a Mersenne Number is far more likely to be prime than any other randomly selected number.

There are also further mysteries surrounding ‘Twin Primes‘. These little buggers are pairs of prime numbers that have a difference of 2. Such as 37 and 39, 3 and 5,
3,756,801,695,685 x 2666689 + 1 and 3,756,801,695,685 x 2666689 – 1…

Are there infinite twin primes? Nobody knows. In fact, the gap between primes excites mathematicians a lot too (they’re very excitable people). The Riemann Hypothesis generalises the distribution of prime numbers, and is possibly the most coveted conundrum in mathematics. Solve it, and have your name written in gold in the world of mathematics forever. When the British mathematician G. H. Hardy went on a dangerous sea voyage in the 1920’s, he sent a note claiming to have proven the Riemann Hypothesis, apparently so that God would not allow him to die.

Surely God would smite him for being a big fat liar?? Anyway…

And so finally we come to the Ulam Spiral. The Ulam spiral is a visual representation of integers, like this:

200px-Ulam_spiral_howto_all_numbers.svg

…which when you look only at the primes, reveals a distinct pattern. Explained nicely (again) by those clever folk over at numberphile in the video clip below (watch full screen):

Are you not fascinated by these bizarre creatures yet? I’m not talking about Dr James Grime, I’m talking about prime numbers.

Are prime numbers useful in society? Sure, Prime Factorisation is the basis of some common methods of cryptology. Essentially, use very large numbers that are the product of two prime numbers (known as a semiprime) as ‘locks’ and their prime factors as ‘keys’.

Something that often stumps people is why ‘1’ isn’t a prime number. Well, it used to be but it was considered so useless (prime factorisation can become infinitely long for example) and required so many special case exclusions, that mathematicians finally succumbed and changed their minds about it.

I’ll leave you with this: There is currently a $250,000 prize for the first person to discover the first billion digit prime. Good luck with that. It at least gives you an indication of the excitement that undiscovered primes cause in mathematicians.

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7 thoughts on “Complements #4 : Prime Numbers

  1. Not about this, but here is my 5th gif ever, still rough !
    And a quote from my post :
    “What use is all this, you ask. well, here is a gif showing that the medians of a triangle are concurrent, and this is preserved under stretch and shear. This means that you only have to prove it for an equilateral triangle. (which is obvious!)”

    • This is my own geometry program, Geostruct, developed slowly over many years, and encouraged by the lack of anything half decent in the 90’s. You can download it from my site mathcomesalive.com. There are numerous ready made and interesting examples, and a help/into file with the basics.
      From what I have seen of GeoGebra (not around in the 90″s) mine is way more flexible and interactive. Ideal for the enthusiastic kid to play with off campus.
      The gif was made from a screen video capture program ScreenCast and an online gif maker. It takes any MP4 video but the free version is limited to 5 seconds (you get 10 frames per second). Site is imgflip.com
      I grabbed the apex of the triangle and moved it while the video was being recorded.

  2. Pingback: Prime Number Tests for 347 | Find the Factors

  3. Pingback: Complements #9 LCM and HCF | Solve My Maths

  4. You can find interesting facts and puzzles about Prime Numbers and Magic Squares, Smith Numbers, and Arithmetic and Palindromic Primes on Glenn Westmore’s blog.

  5. I believe I have stumbled into something very interesting about Prime Numbers along the same lines that you have posted.
    Every prime number greater than 3 is a multiple of 6 + or – 1. I believe that this actually shows that prime number are two separate sequences starting from 5 and 7.
    A) 5 11 17 23 29 35 41 47 53 59 65 71 77 83 89 95 101 107 113 119 125 131 137 143 149… and
    B) 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151…
    All the prime numbers greater than 3 are in this list, but not all the numbers are prime numbers. So how do we identify the composite numbers. I was trying to align the multiples of 5 on a numbers table and the next sequence of 5 6 7 is 35 36 37. What I saw in that table surprised me.
    All of the columns of numbers lined up just like your example, the prime numbers were located on each side of the multiples of 6, all the columns of the multiples of 2 and 3 were grouped together, and 1 column of 6n -1 and another column of 6n +1 were all composite numbers. It seemed that the formula of PN + (PN x 6) could be a way to identify the composite numbers in the list of potential prime numbers. For instance with 5. (5 x 6) would be added to 5 and each following sum.
    35 65 95 125 155
    A second of composite numbers occurred in the column of the square of 5 where the numbers in that column were:
    25 55 85 115 145
    I wondered would it work for 7 as well. 7 x 6 = 42 So the numbers in the first column were:
    7 49 91 133
    Since the square of 7 falls within the first column, the numbers in the second composite column did not seem to be needed.
    So I continued with 11 and added 66 and the sums were:
    77 and 143
    Since the square of 11 falls outside the first column, the second Column of Composite Numbers identified 121.
    13 is where I realized that is on certain calculations you start with the square of the prime.
    So 17 was next, 17 + (17 x 6 = 102) identified that last remaining composite number 119.
    I figured out that the Prime Number Tables that were create by using this formula was creating a systematic process of cross multiplication between the 2 sequences of Possible Prime Numbers (A and B): A x B; A x A; and B x B The second Column of Composite Numbers with the B list (B x A) were all duplicates of A x B.
    So what do you think?
    Here is a video I put on Youtube.

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