If w is an even integer, then w^2 - 1 is not a prime number.

In summary, the conversation discusses the problem of proving that if w is an even integer, then w^2 - 1 is not a prime number. The individual tries to solve it by contradiction, but is unsure of how to proceed. The expert suggests using the fact that w^2 - 1 can be factored as (w+1)(w-1) and notes the exception of w=2 where the result is a prime number. This exception is due to the fact that the integers as a ring have only one unit for multiplication.
  • #1
dgamma3
12
0
hello, I am trying to solve this problem:
If w is an even integer, then w^2 - 1 is not a prime number.

my current working.

prove by contradiction

If w is a even integer then w^2 -1 is a prime number.

if w = 2x
then [itex]w^{2}[/itex] -1
= [itex]4x^{2}[/itex] -1

I am not sure where to go from here, maybe congruence relations:

n+1 = ([itex]4x^{2}[/itex])y
([itex]4x^{2}[/itex])y | n+1 therefore this is a contradiction.

is this correct.
thanks.
 
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  • #2
Don't do it by contradiction, but use:

w²-1=(w+1)(w-1)

also there is the exception of w=2 since then (w+1)(w-1)=3 [itex]\cdot[/itex] 1=3 which is of course a prime. But by splitting it up this way we already see why. either side is the product of primes but this time one side is the empty product. But this is only true if w=2 since the integers as a ring have but one unit for multiplication.
 

1. What does "w is an even integer" mean?

An even integer is a whole number that is divisible by 2 without leaving a remainder. Examples include 2, 4, 6, 8, etc.

2. Why is w^2 - 1 not a prime number if w is an even integer?

When an even integer is squared, the result is always divisible by 4. Therefore, w^2 - 1 can be rewritten as (w-1)(w+1), which means it has at least two factors and cannot be a prime number.

3. Can w be any even integer for this statement to hold true?

Yes, the statement holds true for all even integers.

4. Is there a way to prove this statement using mathematical principles?

Yes, this statement can be proven using the fundamental theorem of arithmetic, which states that every positive integer can be expressed as a unique product of prime numbers. Since w^2 - 1 can be factored into at least two distinct factors, it cannot be a prime number.

5. Are there any exceptions to this statement?

No, this statement holds true for all even integers.

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