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Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p


I have a conjecture:
 Any odd positive number is the sum of 2 to an i-th power and a
(negative) prime.
2n+1 = 2^i+p

for example: 5 = 2+3  9=4+5  15=2^3+7 905=2^12-3191 ....
 as to 2293=2^i +p ,I don't know i , p . it is sure that i>30 000 if
the conjecture is correct.

More,
n = 3^i+p, (if n=6k-2 or n=6k+2)
for example:8 = 3+5  16=3^2+7 100=3+97, 562 = 3^6 -167

I can't proof this. Do you have any idea?


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