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Subject: "Sum of 3 mutually prime numbers"     Previous Topic | Next Topic
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neat_maths
Member since Aug-22-03
 May-19-06, 06:46 AM (EST)
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"Sum of 3 mutually prime numbers"
 
   It'seems possible to prove that the sum of 3 mutually prime numbers x, y and z cannot contain all the primes contained in each of the pairs (x+y), (y+z) and (x+z).
For example x=3, y=4 and z=5 gives (x+y+z)=12 which contains the prime factors in (x+z)=8, (y+z)=9 but not (x+y)=7

Can anyone find a counterexample?

take care


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  Subject     Author     Message Date     ID  
Sum of 3 mutually prime numbers neat_maths May-19-06 TOP
  RE: Sum of 3 mutually prime numbers sfwc May-20-06 1
     RE: Sum of 3 mutually prime numbers neat_maths May-21-06 2
         RE: Sum of 3 mutually prime numbers sfwc May-22-06 3
             RE: Sum of 3 mutually prime numbers erszega May-22-06 4
                 RE: Sum of 3 mutually prime numbers sfwc May-22-06 5
                     RE: Sum of 3 mutually prime numbers neat_maths May-22-06 6
                         RE: Sum of 3 mutually prime numbers sfwc May-23-06 7
                             RE: Sum of 3 mutually prime numbers neat_maths May-24-06 8
                             RE: Sum of 3 mutually prime numbers neat_maths May-27-06 9

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sfwc
Member since Jun-19-03
 May-20-06, 07:33 AM (EST)
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1. "RE: Sum of 3 mutually prime numbers"
In response to message #0
 
   >Can anyone find a counterexample?
x = 1, y = -2, z = 3.

Thankyou

sfwc
<><


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neat_maths
Member since Aug-22-03
 May-21-06, 11:29 PM (EST)
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2. "RE: Sum of 3 mutually prime numbers"
In response to message #1
 
   Good,
Now lets exclude -1, 0 and 1
Take care


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sfwc
Member since Jun-19-03
 May-22-06, 08:07 AM (EST)
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3. "RE: Sum of 3 mutually prime numbers"
In response to message #2
 
   x = -2
y = 3
z = 5

>Good,
>Now lets exclude -1, 0 and 1
This seems like rather arbitrary condition. You have said it 'seems possible to prove' this result. Does this mean that you have an idea for how to prove it? If so, please post that idea here. If not, it is unclear why you imposed this condition; if you wish to impose further conditions please give your reasoning.

Thankyou

sfwc
<><


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erszega
Member since Apr-23-05
 May-22-06, 02:03 PM (EST)
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4. "RE: Sum of 3 mutually prime numbers"
In response to message #3
 
   >x = -2
>y = 3
>z = 5

if we exclude -1, then the choice of x=-2 is not allowed, because a prime number is defined as a positive integer > 1

regards


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sfwc
Member since Jun-19-03
 May-22-06, 06:10 PM (EST)
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5. "RE: Sum of 3 mutually prime numbers"
In response to message #4
 
   x = 3
y = 5
z = 22

>if we exclude -1, then the choice of x=-2 is not allowed,
>because a prime number is defined as a positive integer > 1
I am not sure exactly what you mean by this. Please explain what you understand the condition 'exclude -1, 0 and 1' to mean.

Thankyou

sfwc
<><


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neat_maths
Member since Aug-22-03
 May-22-06, 08:24 PM (EST)
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6. "RE: Sum of 3 mutually prime numbers"
In response to message #5
 
   Thank you sfwc, x=3, y=5 and z=22 is a good counter example.
Back to the drawing board!

Can we change the question to the sum of 3 mutually prime integers greater than 1 containing all the primes in each of the 3 numbers?

For example x=3, y=5 and z=22 their sum (30) contains 2, 3 and 5 but not 11

Take care



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sfwc
Member since Jun-19-03
 May-23-06, 06:45 AM (EST)
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7. "RE: Sum of 3 mutually prime numbers"
In response to message #6
 
   >Can we change the question to the sum of 3 mutually prime
>integers greater than 1 containing all the primes in each of
>the 3 numbers?
x = 8, y = 25, z = 27
If you want to impose both conditions at once I shall need some time to think about it.

Thankyou

sfwc
<><


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neat_maths
Member since Aug-22-03
 May-24-06, 06:22 AM (EST)
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8. "RE: Sum of 3 mutually prime numbers"
In response to message #7
 
   Thank you again swfc!
You guessed it!
It might be possible to prove that no three mutually prime integers greater than 1, say x,y,z can exist where their sum contains all the prime factors in x, y, z, (x+y), (y+z) and (x+z)

take care


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neat_maths
Member since Aug-22-03
 May-27-06, 08:02 PM (EST)
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9. "RE: Sum of 3 mutually prime numbers"
In response to message #7
 
   Please consider this
If the sum of three numbers with no common factor contains the prime factors of each of those numbers or of each pair of numbers then it cannot contain any other prime factors. This is because each pair of numbers is a number itself and their sum is twice the sum of the origional numbers. Two pairs can have a common factor but all three pairs cannot have a common factor.
Your examples
1,2,3 = 1*2*3
(3,5,22)*2 == 8,25,27 = 2*2*3*5

take care


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