Is this phenomenon new?

[spkarra]

Hi,
Here is my statement:
"If k is an positive number, p and q are Prime numbers and p^k NOT = q^k, such that integers of the form (p^k + q^k) will always be an even number ".

I tested with k from 0 thru 99 and for all primes between 1 and 9999. To my surprise, I didnot find a ODD number that satisfies the above condition.
I didnot use Pari/gp.

Just eager to know is there a counterexample for thisphenomenon.
Thanks,
Sastry Karra

****************Mytest results*******************
Run Statistics - Begins ****
Tested for: k = 0 thru 99; all prime numbers between 1 and 9999
Total primes found between 1 and 9999 : 1228
Processing Started at: Thu Jul 23 14:10:35 EDT 2009
Number of integers of the form (p^k + q^k) calculated : 149168844
Number of Even integers of the form (p^k + q^k) 149168844
Number of Odd integers of the form (p^k + q^k) 0
Longest Even number found is: 397 digits long
Processing Ended at: Thu Jul 23 23:42:29 EDT 2009
Run Statistics - Ends ********
**********************************************

[Mini-Geek]

Originally Posted by spkarra

p^k NOT = q^k

This is equivalent to saying k>0 (already established in "k is an positive number") and p!=q (p is not equal to q).

Originally Posted by spkarra

"If k is an positive number, p and q are Prime numbers and p^k NOT = q^k, such that integers of the form (p^k + q^k) will always be an even number ".

What are you saying? I'm going to pretend it says something like "With integer k>0 and two distinct prime numbers p and q, p^k+q^k is always even." Let me know if that's wrong.

Originally Posted by spkarra

I tested with k from 0 thru 99 and for all primes between 1 and 9999. To my surprise, I didnot find a ODD number that satisfies the above condition.

This is not at all surprising, though I doubt the question has been put quite like you have.
Think about it for a second: Will a power of an odd prime number ever contain a factor of two? Will two odd numbers ever sum to an odd number?
The only time that p^k+q^k will ever be odd is if p or q (but not both) is an even prime number: 2.

Originally Posted by spkarra

Total primes found between 1 and 9999 : 1228

There are 1229 primes between 1 and 9999. Unless you've intentionally excluded the possibility of p or q being 2, (which just excluded the only possible time that p^k+q^k is odd) you've got a bug.

[cheesehead]

Originally Posted by spkarra

"If k is an positive number, p and q are Prime numbers and p^k NOT = q^k, such that integers of the form (p^k + q^k) will always be an even number ".

Counterexample:

p = 2, q = 3, k = any positive integer

I tested with k from 0 thru 99 and for all primes between 1 and 9999. To my surprise, I didnot find a ODD number that satisfies the above condition.

I'd have been surprised, too!

"all primes between 1 and 9999" ???

1 < 2 < 3 < 9999

[ATH]

If one prime p or q is 2 but not both then p^k or q^k will be even and the other odd and then (p^k + q^k) wil be odd.

if primes p,q > 2 then p and q is odd and p^k and q^k is odd so (p^k + q^k) will always be even.

[R.D. Silverman]

Originally Posted by spkarra

Hi,
Here is my statement:
"If k is an positive number, p and q are Prime numbers and p^k NOT = q^k, such that integers of the form (p^k + q^k) will always be an even number ".

I tested with k from 0 thru 99 and for all primes between 1 and 9999. To my surprise, I didnot find a ODD number that satisfies the above condition.
I didnot use Pari/gp.

Just eager to know is there a counterexample for thisphenomenon.

Phenomenon? What Phenomenon? If p and q are ANY odd numbers,
their sum is even. This has nothing to do with primes or prime
powers.

Clearly choosing p = 2, and q as any odd number violates your claim.
The sum of an even number and an odd number is odd.

This is grade school arithmetic. Why are you trying to make it into
something mysterious????

[flouran]

Originally Posted by R.D. Silverman

This is grade school arithmetic. Why are you trying to make it into
something mysterious????

Perhaps the OP is in grade school

[Primeinator]

Originally Posted by spkarra

Hi,
Here is my statement:
"If k is an positive number, p and q are Prime numbers and p^k NOT = q^k, such that integers of the form (p^k + q^k) will always be an even number ".

I tested with k from 0 thru 99 and for all primes between 1 and 9999. To my surprise, I didnot find a ODD number that satisfies the above condition.
I didnot use Pari/gp.

Just eager to know is there a counterexample for thisphenomenon.
Thanks,
Sastry Karra

If you think about the basics of exponentiation- any expression x^n will be odd given that x is odd and n is a positive integer. Thus, for p does not equal q... as long as either p or q is odd and the other is even, the ending expression will be odd. Since p and q are prime, the only even prime number is two. Thus, there are an infinite number of counter examples so long as either p or q is equal to two while the other is a prime number > 2.