(This question would be similar to my other on Goldbach's conjecture so I'll change the "rules")
In 1848 de Polignac claimed that "every odd integer is the sum of a prime $p$ and a power of $2$". For example $55=47+2^3=23+2^5.$
Now I have to show that $509$ and $877$ discredit this claim.
What I did is: set $p=509-2^n$ and showed that for every $n$, $p$ is composed (as long as it is nonnegative) checking all the possible values, then I did the same thing for $877$. Fortunately this time I had to check less cases because on the growth of $2^n$ but it still took me a few times.
My question is: how can I prove this without checking case by case and WITHOUT modular arithmetic (which is in the next chap.) ??
EDIT: This question is problem 17 at page 59 (chap.3: the primes and their distribution) of the book "Elementary number theory" by D. M. Burton.