**Defining Prime Numbers Without Numbers**

(Google blog turned out to be very trivial and cumbersome; it is very difficult to make a nice post with many pictures, I wish I knew that before I started it; click here for a pdf version of the post)

In science, a new
representation of already established knowledge may lead to significant
breakthroughs.

One of the great
examples of this happening is the “Feynman’s Diagrams”, invented by Noble Prize Laureate Richard Feynman (obviously). He invented a visual representation of a
complicated mathematical structure of terms of a specific mathematical
sequence, and that invention opened a door into the world of Quantum
Electrodynamics, and then into many other complicated physical worlds.

Every math teacher
knows how to introduce prime numbers. First, one introduces numbers, then how
to add them up, then – a multiplication table, then – a division, factors, and
then – prime numbers as numbers which have no factors except number 1 and
itself (hence, it is divisible only on 1 and itself).

This approach is straightforward,
functional, and effective.

However, for a
math teacher who – for whatever reason – wants to look beyond the standard
strategy, I would like to offer this short piece (and also mention some
reasons).

There is simply no
science without imagination.

Imagination is the
most important ability of a true scientists.

(This picture is borrowed from a large post on the matter)

So, let us imagine
that we have an infinite source of identical squares, “a magic box of squares”,
from which we can take out as many squares as we want to, and they all will be
identical to each other (while reading this post, to defy the deficiencies of Google, also imagine that ALL squares are identical!).
Let’s take a
square and place it on a tabletop.

Now take another
square and also place it on a tabletop. Make the squares be placed close to
each other but not touching each other.

What we see now is
a square near another square.

Now take another
square and also place it on a tabletop. Now make the last square touch one of
the sides of the previous square, so the squares would share a side. When it is
done we see different sets of squares.

Now take another
square and also place it on a tabletop, then another one and use it to make a
set of squares identical to the previous one.

And now take
another square and make the last square touch one of the sides of the previous
square, so the squares would share a side. Arrange the last square in such a
way so you would see a line of squares. When it is done we see again different
sets of squares.

At this point, if
I ask you what type of new sets of squares would we see if we would keep
repeating the action of taking out and placing squares in the same manner as
before, everyone would immediately give an
answer in the form of a picture.

and

and

etc.

There is NOTHING
strange or magical about it.

Our brain is a
very powerful pattern recognition machine.

Our brain can
recognize and produce patterns from the very first day to the very last day of
its existence. A brain recognizes patterns by design, it is its natural
property (which can be advanced by training).

In order to
demonstrate the process of creating sets of squares I had to use many words and
construct complicated sentences. But if a teacher would just show the whole
process to a kid (of a certain age) saying “Look what I am doing, can you do
more?” the kid would be building the sets without any difficulty.

This example shows
that sometimes teaching by doing is much more efficient than teaching by
explaining.

Also, it shows us
that a logical way of reasoning called “by induction” is basically built in
into our brain.

Some people
believe that our brain does ONLY that – recognizes patterns. Those people
believe that if they can build a computer which could recognize patterns as
good as a human brain, they would build “an artificial intelligence”.

Those people are
wrong. A human brain is much much more than just merely a pattern recognition
machine.

But this is a very different discussion (for example, see this link).

But this is a very different discussion (for example, see this link).

However, for every
science teacher the goals of (a) helping students with the development of their
imagination and (b) teaching them how to recognize patterns specific for their
scientific field (the one they teach) are the most important goals of their
teaching practice, more important than drilling students on memorizing and
reciting facts and activities.

OK, let’s return
to our original topic – sets of squares.

Now we have (as we
can imagine it) a set of sets of squares of different sizes.

After learning how
to build as many sets as we want to, we start asking ourselves a natural
question – what can we do with those sets?

Naturally, we can
experimenting as much as we wish, but for the sake of saving time let’s go
directly to the activities we find the most important (i.e. helpful, useful for
us in our everyday practice).

Let us take any
arbitrary set from the sets of our sets of squares, and then again a set from
the sets of our sets of squares (any set – the same as before, or different, it
does not matter) and physically put them together in such a manner so they form
a line of squares.

For example, let’s
take this set

and this set

and place them together
in such a way so their extreme squares would share sides and all the squares
would be on the same level (as we already
know, showing what we want to do may be much easier than describing it by
words, but I strongly recommend teachers to practice in giving a verbal
description of such actions, and even more importantly, asking students to do
the same). The result is obvious, it is this set.

If we compare this
set with the original set of sets, we see that we already have exactly same set
in the set of our sets we have built before.

We can repeat this
action or operation again and again and we will always get the same conclusion;
when we place together (in a specific manner) sets from our original set of
sets, we always get a set from the original set of sets. We can repeat this
action as many times as we need to study various properties of this type of
operation (and develop a science later called “arithmetic”).

At this point it
should become clear for everyone that the language we use for describing what
we do is very much cumbersome. It would make our life much easier if we would
invent a new language, the language which would let us to describe our actions
simpler, clearer, faster (and of course, that language has been developed many
hundreds years ago). The lesson we get from this experience is the
demonstration of how scientists invent new words for describing what they do
(e.g. “a number”, “one”, “two”, “add”, “equal”), and those new words eventually
form a very specific language. A science teacher has to be fluent in that
language and has to teach students that language.

For example, this is the list of the most important terms and categories that every physic teacher has to know in order to teach the first semester of Elementary Physics (borrowed from the syllabus of a full physics course, where everyone can actually read it :) ).

But also a science teacher has to show students

For example, this is the list of the most important terms and categories that every physic teacher has to know in order to teach the first semester of Elementary Physics (borrowed from the syllabus of a full physics course, where everyone can actually read it :) ).

But also a science teacher has to show students

*why*the invention and the use of that language is important (without this linguistic practice there is no science).
Another important
note about language is that a person who does not know the meaning of such
words like “a square”, “a set”, “a line”, “a side”, “put”, “together”, etc.
will not be able to follow our instructions.

Hence, in order to
understand and follow instructions every person needs to have – as a
prerequisite – a certain level of the language development.

One of the very common
reasons for students to fall behind is that the language a teacher uses for
his/her instructions is “foreign” for students.

Second of the very common reasons for students to fall behind is that a teacher does not unveil the logic for making decisions about various aspects of the subject; it's just "do as I say", without "why".

Second of the very common reasons for students to fall behind is that a teacher does not unveil the logic for making decisions about various aspects of the subject; it's just "do as I say", without "why".

OK, let’s return
again to our set of sets of squares and ask again, what else can we do with
them?

Let us take any
arbitrary set from the set of our sets of squares, and then another set from
the sets of our sets of squares (any set – the same as before, or different, it
does not matter) and perform a complicate sequence of steps.

After we select
the sets we physically put them together in such a manner so they do NOT form one
line of squares anymore. Instead, they form an angle. We need to place the extreme
square from any set on the top of an extreme square of another set making sure
that those are the only squares the sets share.

For example, let’s
take this set

and this set

and place them
together as prescribed.

NOW we remove the
square which is placed on a top of another square.

NOW we take a
ruler and a pencil and draw lines which extend extreme sides of the extreme
squares.

We see that the
squares and lines together form a closed figure.

NOW we take from
our “magic” source of squares additional squares and place them in such a way
so they fill in the space inside that closed figure (between the lines and
original squares).

NOW we rearrange
all the squares we have in one line.

Clearly, in the
end we again have a set of squares from the original set of sets of squares.

To summarize,

**we selected a set, then we selected another set, then we performed a complicated operation and in the end we have again a set.**
Now, we can repeat
this type of manipulation as many times as we want to and investigate the
properties of this operation.

One of the
questions we can investigate is what sets from our original set can be produced
as the ending result of such an operation?

To answer this
question we can routinely check all possible sets with all possible sets and
mark all the sets achieved in the result.

We immediately see
that using a square as a set to perform the operation with another set leads to
a trivial result, namely that another set (in a closed figure left after a top
square is removed simply there is no empty space need to be filled).

Hence, we exclude
the trivial case from our consideration.

Further
investigation eventually shows that no matter what sets we use (outside of the
trivial case) there are sets of squares among our original sets which are

*never*get marked. No matter what sets we use (outside of the trivial case), after marking all set we get in the end of the operation,*some sets are never marked*.
Since those sets
are strange, they look special, we give them a special name, we call them “prime
sets”.

The last sentence
concludes our example, we have achieved our goal, we have introduced the notion
of a “prime” mathematical object without using term “number”.

Now we can start
using our advanced mathematical language.

Basically, what we
did we introduced a set of natural numbers. Then we introduced an operation we
call now “addition of natural numbers”. Then we introduced an operation we call
“multiplication of natural numbers”. Note, that those two operations have been
introduced independently, we could have skipped “addition” and go straight to
“multiplication”.

When
“multiplication” is defined, we start multiplying all numbers by each other and
see the result. And we see that some of the numbers from the full set of the
natural numbers

*never*appear as the result of “multiplication” (of course, if we do not use number 1). Those numbers we named “prime numbers”.
It is interesting
to see a relationship between a standard arithmetic representation of numbers
and our “geometric” representation.

For example, if we
take any non-prime number and represent it in a form of a line of squares, we
know that we can always rearrange those squares to from a rectangle. For
example: 15 = 3*5 hence,

we can rearrange
as

If we calculate
the initial and the final

*perimeter*for both figures, we see that in the rectangular form the perimeter is smaller than in the linear form (assuming the “area”, i.e. the number of squares, remains the same).
For example, for
our example, in the linear prom

P = 15*2+2*1 = 32 lines;

in the rectangular
form P = 5*2+3*2=16 lines.

This rule stands
for any non-prime number,

N = M*K:

P

_{linear}= N*2 +1*2 = (N+1)*2 = (M*K+1)*2;
P

_{rectangular}= M*2+K*2 = (M+K)*2.
If M > 1, and K >
1, hence M*K+1 > M+K.

It is easily
proved:

M*K – M > K - 1

M*(K - 1) > K -
1

M > 1

It is interesting
to see that for a non-prime number the rectangular form always provides the
minimum value for the perimeter of the figure formed by a given number of
squares.

However, by the definition
of prime numbers, the figure with the minimum perimeter never has a shape of a
rectangle.

Hence, if we take
a number of squares and try to rearrange it making the figure which has the
minimum value for its perimeter, if that shape is not rectangular, the number
is the prime number.

In other words,

**if from N identical squares we**.*cannot*make a rectangle, number N is prime
This is an
interesting geometrical property of prime numbers.

It would be
interesting to see if this property of prime numbers can be used to quickly
identify if an arbitrary number is a prime number or not (but, since I am not
proficient in the number theory, this quest is outside of my current knowledge).

It also would be
very interesting to find some kind of a physical representation of this process.
For example, a lattice of atoms, with some atoms arranged in a line all in the same
state, and then atoms with the same state start forming a rectangle, keeping
the total number of atoms with the original state the same.