# Thread: Counterexample Help

1. ## Counterexample Help

Find a counterexample to the following assertion:

Every odd number can be expressed as the sum of a power of 2 and a prime.

2. What about 5?

3. 5=2+3... and 3 is prime...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. Originally Posted by MATNTRNG
Find a counterexample to the following assertion:

Every odd number can be expressed as the sum of a power of 2 and a prime.
Haha, 1.

5. Originally Posted by chisigma
5=2+3... and 3 is prime...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
Haha I forgot about $\displaystyle 2^1$.

But yeah, 1 would work

6. The smallest odd number > 1 that can't be written as the sum of a power of 2 and a prime number is 127...

127-1= 126= 2 x 7 x 9

127-2 = 125 = 5 x 5 x 5

127-4 = 123 = 3 x 41

127 - 8 = 119 = 7 x 17

127 - 16 = 111 = 3 x 37

127 - 32 = 95 = 5 x 19

127 - 64 = 63 = 3 x 3 x 7

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

7. There are seventeen odd numbers >1 and <1000 that can't be written as the sum of a power of 2 and a prime number:

127,149,251,331,337,373,509,599,701,757,809,877,90 5,907,959,977,997...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

8. Originally Posted by MATNTRNG
Find a counterexample to the following assertion:

Every odd number can be expressed as the sum of a power of 2 and a prime.
If you're only allowing positive numbers and addition, then 1 and 3 would fill that bill.