Expert: Scott A Wilson - 4/27/2009

Question
QUESTION: Why isn't "1" considered a prime number? According to wikipedia the numbers that are multiplied together must be distinct, and one of the numbers being 1, and the other being the number in question. Another source says that "1" is in fact a prime number, but is rarely considered as such because it complicates some applications. What purposes do prime numbers have? Thank you.

ANSWER: You hit on a hot topic of mine.

As I learned it, the best way to describe prime numbers is that
they have two distinct factors - 1 and themself.  This excludes 1 since it only has 1 factor.  That is the only number with this property and in most cases it is best to not consider it as a prime number.

The number two is known as the first prime and is also the only even prime (since all of the rest of the evens have two as a factor).  The number two is frequently used on computers which operate at the ground leve in base two, which is also known as binary.  To you and me, 127 is 127, but to the computer is it 1111111.  
That is, 1(2^6) + 1(2^5) +1(2^4) + 1(2^3) + 1(2^2) + 1(2^1) + 1(2^0).
That's 64+32+16+8+4+2+1.

But back to the subject.  Very high prime numbers are frequently used in codes by the govenment.  In people who crack the code must know the prime number involved.

As far as primes go, prime numbers always have 2 distinct factors, so one is not included since it has only 1 distinct factor.  When a prime is squared, it has three factors.  (Ex. 25=5², factors are 1, 5, 25).  When taken to the fourth, the have 5 prime factors (Ex. 7^4=2401; factors are 1, 7, 49, 343, 2401 }.  What should be noted here is that the only numbers with a prime number of factors are prime numbers to a prime - 1.  The two primes can be different and the second prime is the number of factors.  For example, since 2 and 11 are both prime, 2^(11-1) has 11 factors.  They are 1, 2, 4, 8, 16,32, 64, 128, 256, 512, and 1024.  Since 3 and 7 are prime, 3^(7-1) =3^6 = 729.  The 7 factors are 1, 3, 9, 27, 81, 243, and 729.  This is another reason not to think of 1 as a prime.  Ex: 6 is not prime, but 6^(1-1) = 6^0 = 1, which only has 1 factor, therefore don't think of 1 as a prime or the theory of prime^(prime-1) does not hold.

For example, take 1700.  The factors are 2*2*5*5*17.  Since there are 2-2's, 2-5's, and 1-17, take (2+1)(2+1)(1+1) = 3*3*2 = 18.  That number has 18 factors.  The number 1 only has 1 factor and is therefore not considered prime.  

Another property of prime numbers is that they always increase the number when multiplied.  Therefore 1 can't be a prime.

When listing out the prime factors, we might saw that 112 = 2*2*2*2*7, which is a list of the prime factors.  If one was considered as a prime number, we could say the list of prime included a 1, or 2-1's, or 3-1's, or 1008-1's, or as many 1's as we felt like.  That's why 1 is not considered as a prime number.

Now 1 to any power is 1.  That creates a very boring sequence of numbers.  That might not be a very good reason, but prime numbers give a much more interesting sequence of numbers.  For example
2 gives 2, 4, 8, 16, 32, 64, 128, 256, 512, ....
3 give 3, 9, 27, 81, 243, 729, ...
13 gives 13; 169; 2197; 28,561; 371,293; 4,826,809; ...

By the way, 729 is a very interesting number.
729 = 3^6, 9^3, or 27^2.  That is because it is to the 6th.
Also, 1,729 is the smallest sum of two cubes.
1,729 = 9^3 + 10^3 and 1,729 = 1^3 + 12^3.

Any number that is p^6 is also (p^2)^3 and (p^3)^2.
Examples are 64 (2^6, 4^3, 8^3).  
15,625 = 5^6, 25^3, 125^2 where 25=5^2 and 125=5^3.
117649=7^6, so it is also 49^3 or 343^2 since 49=7^2 and 343=7^3.

What can I say?  Numbers fascinate me.


---------- FOLLOW-UP ----------

QUESTION: Although you've given me many reasons why it is inconvenient for 1 to be a prime number, I am interested stickly in the technicality of it. Is 1 under any circumstances prime? And has the definition ever changed? And lastly on a different topic how does one find square routes long hand? I was never taught that in high school or college.

Answer
To say it now, 1 is not considered a prime number.

Let us calculate the squareroot of 77284.
First, starting at the decimal, divide the number is sets of two digits.  That would give us first an 84, then a 72.  To write the number in this format, it is 7 72 84.

The first thing to do is to look for the biggest square that is less than the first digit.  1²=1, and that's less than 7.  2²=4, and that is less than 7.  3²=9, and that is not less than 7.  So we know the number starts with a 2.  Since there are 3 groups we divided the number into, we know that the squareroot has 3 digits.  From this, we know that the squareroot is two hundred and something.  Now if the square of 200 is looked at, it can be seen to be 40,00, which is less than our 77,284.

Once this number has been found, it is the first digit.  We then need to square it and subtract it from the 7 in front, giving 3.  We then bring down the next two digits.  This gives us 372.  Now note that the first digit was 2.  Multiply the digits gotten by 2 and a blank at the end.  What we are looking for is 4_ x _ to be less than 372.

Lets try 6.  46 x 6 = 276, and that is less than 372.
Lets try 7.  47 x 7 = 329, and that is less than 372.
Lets try 8.  48 x 8 = 384, and that is not less than 372.
Since it's not less, back up one.  The 7 was right.
Put that up top right after the 2.  We now have 27__.
Subtracting the 329 from the 372 gives us 43.

Take the 43 and bring down the next two digits, so we have 4384.
Take the number up to and multiply by 2, giving 27 x 2 = 54.
Find the number to fill in the blank.
54__ x __ < 4384.
Lets try 7.  547 x 7 = 3829, and that is less than 4384.
Lets try 8.  548 x 8 = 4384.  Hey!  
That is 4384, so we know that 77,284 is a perfect square.
Looking at the numbers up top, we had a 27, and now we add 8.
That gives us 278.

Now the way to square numbers is in your head.
You only have to know the square of the first 25 numbers.
With that, you can determine the squares of all numbers.
The first thing to do is find the closets 50.
That would be 300.  Now 278 is 300-22.
So when we square (300-22), we get (300-22)².  
That wasn't hard, but I'm not finisned yet.

Remember algebra, (a+b)² = a² + 2ab + b²?  Well, lets use it.
Now we know that 3² is 9, so 300² is 9 with 4 zeros, or 90,000.
Now I know that 22² is 484, so now we have 90,484.
We know that a is 3 and b is 22, so 2ab is 2*3*22=132.
Now note the 3 is really for 300, so the answer is 13,200.
Note that also it is a -22, so that makes it -13,200.
Now, what was our number?  Oh yeah - 90,144.

Now to subtract 13,200, giving, well, 90-13 is 77,
so at first we have 77,484, but we still need to subtract 200.
4-2=2, soo 400-200=200, so the number is now 77,284.
Looking back up near the top, that's what I started with.