# Reducing numbers

There is something going through my mind recently.
is it possible to reduce any number to the following form:
x^n +y , and -x+1<y<x-1?? x isn't necessarily prime
or better something like this x^n +y where y=+1 or -1
I tried many numbers, so far I can't see a contradiction to these 2 rules I stated, or maybe there is one, but can't really see it.
I want to see if it's possible to reduce a very very big number, to a simpler form, like the 2 i stated above.
example:(I don't know if this is correct)
1...million zero...1
it should be reduced to this 1000^1000 +1

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Homework Helper
One assume you do not consider n=1 acceptable.

If not then you obviously can't have the extra condition that y=+/-1, as not every number is one more or less than a perfect power.

what about the first way I thought of ?
x^n +y , and -x+1<y<x-1 not possible too ?

Homework Helper
Again, you're assuming n>1, obviosuly, but not bothering to state it. And again it is trivial to show it is possible if you relax it to less than or equal in the condition with y. And always with n=2. What are the two extremes? x(x-1) and x(x+1).

I leave it to you to finish that proof, and to think what it implies for your other question with a strict inequality.