Question

What properties do prime numbers exhibit which can be used in proofs to define them?
Like rational numbers have a unique property that they can be expressed as a quotient of a/b.
Even numbers have a unique property of divisibility by 2 and thus they can be expressed as 2x.
Similarly are there any unique properties for prime numbers?

Answer 1

@ganeshie8

Answer 2

I guess you could say if two integers multiply together to equal it one has to be one and the other has to be the number, or to put it another way if a*b=c and both a and b are integers that aren't 1 then c is not a prime.

Answer 3

Like if there is a question
"Prove that a prime number......"
For this I have to assume there is a prime number x
How should I define this x in mathematical terms ?

Answer 4

To be honest that is about it, I don't suppose you have a specific proof in mind.

Answer 5

Prove that there prime numbers beyond 3 are of the form 6x+1 or 6x-1

Answer 6

That is true you could say it is not divisible by anything but one and itself

Answer 7

How could I say that
X is a prime number such that X =1*X
I can't use the same thing in its definition
Its like saying "Prime number is a number such that it is prime"

Answer 8

you could say x is a prime number such that x/y is not an integer when y is not 1 or x.

Answer 9

I cant use x in the definition that's the problem.
Its like saying X is a prime number such that a PRIME divided by !(1 or itself) is an integer
The contradiction is what is the PRIME?

Answer 10

@Loser66 I dont need the solution to that proof. I want a way to define a prime number

Answer 11

I think only one way is to find its factor

Answer 12

@ganeshie8

Answer 13

Hey!

Answer 14

Hey

Answer 15

Stick to the definition of a prime.
By definition, a prime number is an integer whose positive factors are 1 and itself.

Answer 16

So, in order to prove a number is prime, you just have to prove that the only positive factors of it are 1 and itself.

Answer 17

And how can I do that?

Answer 18

Consider the example :

Prove 7 is a prime

Answer 19

7 is not divisible by 2, 3, 4, 5, 6

Answer 20

therefore it is a prime. end of story.

Answer 21

Prove that 6x+1 is prime if 6x+1>3. How should I use your way here

Answer 22

That statement is wrong

Answer 23

6*4 + 1 = 25 is not a prime

Answer 24

Oh sorry 6x+1 or 6x-1

Answer 25

That should also be wrong.

Answer 26

Can you put the full statement in one reply ?

Answer 27

Prove that there prime numbers beyond 3 are of the form 6x+1 or 6x-1

Answer 28

That looks good. Do you see yet why I say your earlier statements are wrong ?

Answer 29

Yes, the implying relations were wrong. Here the converse of this statement is not true

Answer 30

Every prime number beyond 3 is of form 6x + 1 or 6x - 1

is not same as saying

Every number of form 6x + 1 or 6x - 1 is a prime

Answer 31

Good.

Answer 32

Let me ask you a question

Answer 33

When you divide a number by 6, what are the remainders you expect ?

Answer 34

smaller than 6

Answer 35

What exactly are they ? Can you list them here ?

Answer 36

1,2,3,4,5

Answer 37

You missed 0

Answer 38

Oh yes

Answer 39

0,1,2,3,4,5

Answer 40

Would you agree that ANY integer can be expressed as 6x + k,
where x is some integer and 0 <= k < 6 ?

Answer 41

Obviously

Answer 42

Sure ?

Answer 43

yes

Answer 44

What are x and k for 25 ?

Answer 45

x=4 k=1

Answer 46

Yes. How about 30

Answer 47

x=5 k=0

Answer 48

oh got it done.

Answer 49

Good. So we can express every integer in the form 6x + k,
where x is some integer and k is between 0 and 6

Answer 50

k cant be 2,3,4 but it can be 1 or5 which is in other words 6-1 thus the only established values for k are 1 and -1 for a prime

Answer 51

why can't k be 2,3,4 ?

Answer 52

well multiple of 6x + 2 divisible by 2
6x+3 divisble by 3
6x+4 divisble by 2

Answer 53

You got it!

Answer 54

But I am stuck at my question. How should I define a prime
If say I assume that X is a prime such that .....(what should I write here)?

Answer 55

As I said, stick to the definition. There is no known form for a prime. They are strange and hard.

Answer 56

If you assume x is a prime, then the definition says that the only factors of x are 1 and x.

Answer 57

Can't I just use the 6x+1 or 6x-1 definition?

Answer 58

You may use it. Just be cautious that not all integers of form 6x + 1 or 6x-1 are primes.

Answer 59

As you said, the converse of a statement need not follow the truth value of the actual statement.

All horses have four legs

doesn't imply

All four legged animals are horses

Answer 60

Okay one more
Prove by contradiction that if x<25 and y<25 then x+y>50 is false

Answer 61

I know the basic idea. But How should I express it

Answer 62

Any contradiction proof starts by assuming the opposite of what needs to be proven.

Answer 63

So start by assuming the opposite of whatever you want to prove.

Answer 64

x+y >50 is true

Answer 65

Yes

Answer 66

x<25
y<25
x+y<25+25
x+y<50 Right???

Answer 67

That looks okay, but where's the contradiction ?

Answer 68

x+y<50 since we assume x+y>50 is true.
But my main question is it okay to add or subtract inequalities like that. Are there some laws that govern their manipulation?

Answer 69

It is perfectly okay if it makes sense to you.

Answer 70

x < 25
y < 25

do imply x + y < 25 + 25

I don't see anything wrong here. Do you ?

Answer 71

Two sticks that are less then 25 cannot add up to a length more than 25+25.

Answer 72

Easy to see right ?

Answer 73

yes, but what if I am squaring or applying a function on them or multiplying by something
like x^2 * y^2 <2500*2500. That would be correct right?

Answer 74

Looks good to me. But keep in mind, each case is different. There is no one single trick that works in all cases. If you followed the rules of math correctly while manipulation, you would be fien.

Answer 75

Oh okay, one thing. Are you always online at this time?

Answer 76

Mostly yes. If not, you may tag me in your question and I'll respond when I login. I like your questions :)

Answer 77

See it gets a little burden for me be online for the whole day to get help from you hoping that you would come online at one moment or other
Thanks

Answer 78

inequalities become a bit tricky when you allow negative numbers

x < a

need not imply

x^2 < a^2

Answer 79

Yeah I learnt that. We have to account for their increasing and decreasing properties

Answer 80

-5 < 1 is true
but

(-5)^2 < 1^2 isn't

Answer 81

So you must use your reason when manipulating these. Simply squaring both sides doesn't work. You may add or subtract stuff both sides though.

Answer 82

Okay one more thing
in proof in which I have to deal with digits of integers how should I do them? Like prove that number whose digits add up to 3x is divisible by 3 (I made this question so it isn't necessary for it to be true.)

Answer 83

let me ask you a question

Answer 84

what does the number 435 represent ?

Answer 85

4 hundreds 3 tens 5 ones ?

Answer 86

Exactly!

435 is a shortcut form for the decimal expansion :

4*10^2 + 3*10^1 + 5*10^0

Answer 87

4*10^2 + 3*10^1 + 5*10^0 looks lengthy and ugly

so we express it in compact form as 435

Answer 88

Now look at the powers of 10

Answer 89

Okay

Answer 90

What is the remainder when 10 is divided by 3 ?

Answer 91

1

Answer 92

What is the remainder when 10^2 is divided by 3 ?

Answer 93

it will remain 1 if n>0

Answer 94

Good. So any power of 10 leaves the remainder 1 when divided by 3

Answer 95

Look at our number 435 :

4*10^2 + 3*10^1 + 5*10^0

Answer 96

As far as remainders when divided by 3 are concerned, can we say the number is equivalent to :

4*1^2 + 3*1^1 + 5*1^0

?

Answer 97

That is same as

4 + 3 + 5