Two and three - the first two primes.

power(0) is (: x->1 :), power(1+n) is (: x-> x.power(n,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) = (: x-> power(n,x).power(m,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.