# A Poster About Prime Numbers [closed]

We're going to design a poster about prime numbers, which will appear in a mathematics magazine for middle school students.

The poster should be both visually attractive and mathematically rich.

Do you have any suggestion for the content or idea of the poster?

Thanks.

Update: Finally! Thanks to the ideas offered here, a poster (based on the sieve of Eratosthenes) was designed and printed as the cover of our magazine. The numbers below the sieve are $4, 6, 8, 9, 10, 12$. The numbers inside the sieve are $2, 3, 5, 7, 11$. The number above the sieve is $1$.

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## closed as primarily opinion-based by apnorton, ml0105, M Turgeon, RecklessReckoner, heropupApr 22 '14 at 3:24

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

Perhaps Eratosthenes' Sieve? murderousmaths.co.uk/books/MMoE/erat.htm – Prahlad Vaidyanathan Apr 21 '14 at 15:59
@PrahladVaidyanathan Thanks. Yes, it's mathematically rich but unfortunately is not visually attractive. By the way, it would not be new to the students; they encounter it in middle school math textbook. – Behzad Apr 21 '14 at 16:02
@Behzad,it is if you use lots of colours. – rah4927 Apr 21 '14 at 16:11

About a week ago I have been working on visualising primes on Ulam's spiral and different variations. These pictures are for first $100 000$ natural numbers, while primes are colored with blue. It could work as a pattern behind text. Or you can create it for say 1000 numbers and add also the numbers (with bigger squares, so it is readable). Also you can highlight primes generated by some lets say quadratic form $x^2 + bx + c$. It is math, possibilities are endless. Have a look:

Triangle:

Hexagon:

Rectangle:

Square(standard Ulam spiral):

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Very neat! Since it took me a minute to realize, I'll note for anyone else looking that the very strong horizontal patterns in the left rhombus of the hexagon come about because values there (but not so cleanly along axis lines in other 'sextants' of the spiral) differ by a multiple of 6, so they've effectively been pre-sieved for divisibility by 2 and 3. – Steven Stadnicki Apr 21 '14 at 16:33

Gaussian primes from http://mathworld.wolfram.com/GaussianPrime.html.

Toss [that]/[something similar], in, maybe in the corner of the poster or somewhere subtle, just for fun.

I like Gaussian Primes personally.

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I think something that shows what makes a number not prime would be interesting. You could have a list of numbers (2, 3, 4, etc) filling the poster. Primes would be a particular colour, say white. Each prime would have a border around it with a unique colour, such as blue for 2, yellow for 3, etc.

A number that isn't prime would be coloured with the border colours of the primes that compose it. In this example the number 8 would be blue. If a number has two or more prime factors it could be partially each colour. In this example 6 would be half blue and half yellow.

You could also colour the composite numbers as a combination of its factors, making 6 green. But that'd probably be less intuitive and might end up with some weird shade of brown down the line.

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A friend of mine once wore a t-shirt with this idea for the naturals up to 100 I believe. Something like this – Dan Rust Apr 21 '14 at 16:08
I am sure I saw an answer on MSE bearing the exact same idea. – rah4927 Apr 21 '14 at 16:12
@EuYu,indeed,that's what I was talking about. – rah4927 Apr 21 '14 at 16:30

This idea requires a person who has some talent for art. I don't have it.

Eratosthenes could be appear holding a sieve (a traditional one). Falling from the sieve could be some composite numbers. Inside the sieve, if possible (perhaps it would be difficult to draw and to see) would be some primes. The number one is observing the process with curuosity. In fact, he (?) is scratching his head.

Regardless of the final poster, I wish you luck.

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I would rather like to see this! :-) – Marius Kempe Apr 21 '14 at 16:23
Cute, but I'm not sure it fits the purposes of the question - if it's supposed to be "mathematically rich", I think something like the Ulam spiral would be better. – Jack M Apr 21 '14 at 19:59

Well, you can draw some numbers in the foreground, starting from $2,3$ and $5$, talking together, like in a comic strip...i.e

• $2:$Hi $3$! ... Oh, come on! why are you so sad?
• $3:$Hi 2! nothing..it's always the same old story: I've invited many of my friends to my birthday party, but almost none of them came: they were so far...I love you $2,5$, but I'd like to meet new people, experience new things..you know..
• $5:$ oh, $3$, don't make it so terrible! Look! I've brought with me my other twin, $7$!
• $7:$ Hi guys! Don't you mind if I came with some of my friends? Do you?..it'just that $11$ and $13$ are bounded like brothers, $17$ won't separate himself from $19$, and $23$ is in love with $29$, who can't go to a party without $31$: you know, they are old twins, and now they trust each other like they were little kids...
• $11,13:$ Hi everyone! Look! We're so many!
• $17,19:$ Yes, of course! We, prime numbers are less than those bad boys of the gang the multiple of 2, or the multiple of 3, but we're special! None can divide us! We'll stay together forever!
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My daughter (age 13) said tell me what primes are and why they are useful. Maybe have some text boxes in the shape of primes.

I thought some of the following:

what we know - an infinite number of primes, unique prime factorisation for integers, maybe little Fermat, Dirichlet infinite number of primes in AP (and perhaps the Green Tao Theorem and why they are different), how to estimate the number of primes quite accurately

What we don't know - how to find the prime factors of a number efficiently, Goldbach conjecture, how to find the next prime number without a lot of work

Uses: eg coding, design of gear wheels

What we are learning: Eg the Zhang/Maynard/Tao etc work on the gaps between primes - see work in action on the internet.

How to join in: Prove or disprove the Riemann Hypothesis - win £££/$. Join GIMPS. - I'll add another different suggestion to that of my first if you'll allow. What about some sort of timeline of primes discovered with some nice caricatures of how they might have been discovered? As user ajotatxe remarked: This idea requires a person who has some talent for art. I don't have it. This also applies to me but I'll give it a shot. For example, you might initially have some people counting with their hands, or scribbling on paper, (actually show a few prime numbers), maybe even a sort of sieve like others suggested but I wouldn't want to rob their ideas, or some feint marks resembling modular calculations. A few random guesses for the Mersenne Numbers which were thought to be prime, question marks and maybe even one or two of the smaller ones with question marks. Maybe move onto some inkling of Lucas and Lehmer's method and/or others, which I don't claim to have read/be any sort of expert in. You can top it off with an effective "comical" drawing of asking a computer, is such a number prime, and it say "true", maybe even with an implied "ding" just for effect :) - This is just an idea for an element to use. I think the poster might show the multiplication table modulo some prime number, and for contrast the multiplication table modulo a (highly) composite number. If space allows I would suggest$23$and$24$. Looking carefully tables will look quite different, for instance modulo a prime number each row and each column (except the one of$0\$) contain every possible number once, but this fails miserably for some (often many) rows of the table modulo a composite number. The observation tha multiplication modulo a prime behaves very specially (technically, it defines a field) I think goes to the heart of why mathematicians are interested in prime numbers in the first place. (The mysteries of the distribution of prime numbers are certainly deep and intriguing, but would be a mere curiosity is the notion of prime number would not be of such a fundamental importance in the first place.) It is this property that makes (large) prime numbers important in cryptography for instance (together with the difficulty of actually finding prime factors, but that is something that I would not know very well how to visualise).

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As several posters suggest, some sort of Eratosthenes idea is the obvious way to go. So perhaps an array of numbers with one type of composite picked out in one colour and removed in the next.

1st row, all the natural numbers ;

2nd row, all the odd natural numbers ;

3rd row is 2nd row with composites dividing by 3 removed ;

4th row is 3rd row with composites dividing by 5 removed....

...keeping all numbers in the same columns as in the 1st row, so everyone can see the composites steadily being thinned out until only primes remain, still in their original positions.

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