TNT (Typographical Number Theory)

Wald's Solution for "A is a Power of Ten"

The following full solution to the "power of ten" problem is from Kevin Wald:

Well, one summer, I decided to tackle the problem. Not having any great knowledge of number theory, I used a more brute force approach. Note that for greater comprehensibility, I have broken the resulting formula up into several stages, but it would not be difficult to put it back together into one vast formula:

{e is prime:} PRIME(e) := ~Eb:Ec:((b+2)*(c+2) = e) {if e is a prime, true iff a is a power of e:} PPOWER(a,e) := Ab:Ac:((b*c = a) -> ((b = 1) or (Ed: (e*d) = b))) {if e is a prime, and a is a power of e, true iff d is the (log_e a)th digit (counting from the right, starting with 0) in the base-e expansion of n:} DIG(a,e,d,n) := (d < e) & Eb:Ec:((c < a) & (n = (b*e*a) + (d*a) + c)) {if e is prime, and a is a power of e, true iff n has exactly (log_e a) digits in its base-e expansion (0 is counted as having 0 digits:} LENGTH(e,a,n):= (n < a) & Ab:((PPOWER(b,e) & (b < a)) -> (b <= n)) POWER_OF_TEN(x):= Ee:(PRIME(e) & (e > x) & En:Ea:(LENGTH(e,a,n) & DIG(1,e,1,n) & Ai:Aj:Au:( (PPOWER(u,e) & ((e*u) < a) & DIG(u,e,i,n) & DIG(e*u,e,j,n)) -> j = (10 * i) ) & Eu:(PPOWER(u,e) & (e*u = a) & DIG(u,e,x,n)) ) )

The basic idea is that you are asserting that for some prime e greater than x, we can find a number n which, when expressed in base-e notation, will have 1 in its units place, 10 in its e's place, 100 in its (e^2)'s place, and in general have the "digit" in each place be 10 times greater than the one to its right, such that the leftmost digit is our x.

To translate into Hofstadter's notation, of course, we must use horse-shoes instead of ->'s, big carets instead of &'s, letters a through e (followed by however many ''s) exclusively, and so forth. (We must also replace <'s and <= with appropriate expansions, and substitute in for our capitalized formula abbreviations.) This is left as an exercise to the reader.

- Kevin Wald (to rec.puzzles), 1993 Mar 12

Here are the values of the smallest suitable prime e for the first few powers of 10, and the resulting value of the number n that encodes the sequnce {1, 10, 100, ..., x}:

	x	e	n
0	1	2	1
1	10	11	1+10×11 = 111
2	100	101	1+10×101+100×101² = 1021111
3	1000	1009	1+10×1009+100×1009²+1000×1009³ = 1027345547191
4	10000	10007	100281296248708457971
5	100000	100003	1000151009121275495160786137931
6	1000000	1000003	1000018100136510549122242550599608364137931
n	10ⁿ	~ 10ⁿ	~ 10^n²

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2013 Feb 26.

s.27