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7:33 AM
$\lceil \log_b (x+1) \rceil = \lfloor \log_b (x) \rfloor +1$ right?
 
 
1 hour later…
8:44 AM
@MrPie Not for all mentioned $n$
@Mathphile The polynomial is irreducible for the following $n\le 500$ :
gp > for(n=1,500,if(polisirreducible(x^(2*n)-x^n+1)==1,print1(n," ")))
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192 216 243 256 288 324 384 432 486
gp >
Can it be that it is irreducible for $n>1$ if and only if $n$ has no prime factors except $2$ and $3$ ?
 
 
1 hour later…
10:22 AM
@Peter This answer proves your observation
@Mathphile Consider $x=b^{1-10^{-k}}$ for large $k$
 
10:38 AM
@TheSimpliFire Nice, so I guessed correctly !
I still would need help to show the compositeness of various numbers of the form $$2020^{2^n}+1$$
 

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