power(0) is (: 1 ←x :), power(1+n) is (: x.power(n, x) ←x :),
for each natural n (x varies over whatever values we know how to multiply by
themselves and 1). One may equally define: power(1) is the identity,
power(n+m) = (: power(n,x).power(m,x) ←x :) and leave power(0) out of the
discussion when no multiplicative identity is available.
Notice, for natural i, that power(power(i, 3), 2) +1 is always a multiple
of power(i+1, 3). Proof:
For i = 0,
power(power(0,3),2) is power(1,2) is 2 and 2+1 is 3 which is power(0+1,3);
once known for i−1, so i>0 and 2.i>i, making
power(power(i−1,3),2) equal to q.power(i,3)−1 for some natural q,
we can cube each to find power(power(i,3),2) equal to the cube of
q.power(i,3)−1;
which comes as a series of terms, in steadily diminishing (from 3)
powers of q.power(i,3);
given 2.i>i, all terms but the last two (powers 1 and 0) are
multiples of power(2.i,3) hence, in particular, of power(i+1,3);
furthermore, the coefficient of the power(1) term is 3, making it also a
multiple of power(i+1,3);
leaving power(power(i,3),2)+1 equal to some multiple of power(i+1,3);
QED.
Now I'm more interested in a power of 3 that is one short of a power of
2. The obvious example is 3 itself, one short of 4, but is there a bigger ?
Lest you ask why, powers of 3 provide an interesting number system, which I'd
ideally encode within the conventional binary context leaving as little slack
as possible – but just enough to (for example) distinguish keep
reading from that was the last digit of the number. For these
purposes, 243 = power(5,3) and 256 = power(8,2) differ by only 13, which is
pretty good: a byte can encode a bundle of 5 trits or an out-of-band
message.
So let me look at polynomials in the powers of 2 and 3; and I'll
abbreviate power as just p.
I'm also interested in which powers of 2 are one more or less than a
prime. Conjecture: every prime is the sum or difference of a power of 2 and a
power of 3.