## "Four"ier transform

I liked the joke and am familiar enough with the math of working in unusual bases that I felt a need to implement a quick version of this in Python. Code follows.

 `````` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 `````` ``````#!/usr/bin/env python def fourier(x, b): """Attempts to find a fourier version of x, working down from base b. Returns the fouriest base.""" mostFours = 0 bestBase = -1 for base in range(b, 1, -1): fours = 0 t = x while t != 0: if (t % base) == 4: fours += 1 t //= base # Prefer lower bases if fours >= mostFours: print(baseconvert(x, base) + "_{0}".format(base)) mostFours = fours bestBase = base return bestBase BASE_CHARS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ" def baseconvert(x, base): s = "" while x != 0: s += BASE_CHARS[x % base] x //= base return ''.join(reversed(s)) if __name__ == '__main__': from sys import argv, exit if len(argv) < 2: print("""Usage: {0} Computes the "four"ier transform of , printing the optimizations to reach the "fouriest" form.""".format(argv)) exit(1) x = int(argv) # Base 36 is the largest sensible base to use base = fourier(x, 36) if base == -1: print("{0} is four-prime!".format(x)) ``````

This is Python 3.x code, using explicit integer division. It should work under the 2.x series if you change line 34 to use “/=” rather than “//=”. It can only go up to base 36, because I didn’t want to deal with bases that are hard to represent in reasonable ways. Up to base 64 is an option, but in that case I would have wanted to use MIME base 64, which puts digits at positions 52 through 61, which would be confusing to read. Thus it only supports up to base 36, but could be adjusted with relative east to do larger bases.

Running a few examples:

 `````` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 `````` ``````\$ python fourier.py 624 HC_36 HT_35 IC_34 IU_33 JG_32 K4_31 143_23 1B4_20 440_12 4444_5 \$ python fourier.py 65535 1EKF_36 1IHF_35 1MNH_34 1R5U_33 1VVV_32 2661_31 2COF_30 2JQO_29 2RGF_28 38O6_27 3IOF_26 44LA_25 B44F_18 14640_15 4044120_5 \$ python fourier.py 3 3 is four-prime! ``````

A few quirks: it prefers lower bases, so bases that match earlier attempts in fouriness will be printed, despite having equal fouriness. I’ve decided to call values that have no representations containing a ‘4’ character “four-prime”, which is probably going to be a rare occurrence, but the program handles it okay.

Generalization of the algorithm is certainly possible, and basically requires changing the condition on line 14 to match different digit values. For example, a hypothetical “Three"ier transform would replace the ‘4’ on line 14 with a ‘3’.