## The Beginning of the Number Pi, in Binary Through Hexadecimal, etc.

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Binary (base-2) pi: 11.00100 10000 11111 10110 10101 00010 00100 00101 10100 01100 00100 01101 00110 00100 11000 11001 10001 01000 10111 00000 . . .

Trinary (base-3) pi: 10.01021 10122 22010 21100 21111 10221 22222 01112 01212 12120 01211 00100 10122 20222 12012 01211 12101 21011 20022 01202 . . .

Quaternary (base-4) pi: 3.02100 33312 22202 02011 22030 02031 03010 30121 20220 23200 03130 01303 10102 21000 21032 00202 02212 13303 01310 00020 . . .

Quinary (base-5) pi: 3.03232 21430 33432 41124 12240 41402 31421 11430 20310 02200 34441 32211 01040 33213 44004 32444 01441 04233 41330 11323 . . .

Heximal (base-6) pi: 3.05033 00514 15124 10523 44140 53125 32110 23012 14442 00411 52525 53314 20333 13113 55351 31233 45533 41001 51543 44401 . . .

Septenary (base-7) pi: 3.06636 51432 03613 41102 63402 24465 22266 43520 65024 01554 43215 42643 10251 61154 56522 00026 22436 10330 14432 33631 . . .

Octal (base-8) pi: 3.11037 55242 10264 30215 14230 63050 56006 70163 21122 01116 02105 14763 07200 20273 72461 66116 33104 50512 02074 61615 . . .

Nonary (base-9) pi: 3.12418 81240 74427 88645 17776 17310 35828 51654 53534 62652 30112 63214 50283 86403 43541 63303 08678 13278 71588 . . .

Decimal (base-10) pi: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 . . .

Undecimal (base-11) pi: 3.16150 70286 5A48 . . .

Duodecimal (base-12) pi: 3.18480 9493B 91866 4573A 6211B B1515 51A05 72929 0A780 9A492 74214 0A60A 55256 A0661 A0375 3A3AA 54805 64688 0181A 36830 . . .

Tridecimal (base 13) pi: 3.1AC10 49052 A2C7 . . .

Tetradecimal (base-14) pi: 3.1DA75 CDA81 3752 . . .

Pentadecimal (base-15) pi: 3.21CD1 DC46C 2B7A . . .

Hexadecimal (base-16) pi: 3.243F6 A8885 A308D 31319 8A2E0 37073 44A40 93822 299F3 1D008 2EFA9 8EC4E 6C894 52821 E638D 01377 BE546 6CF34 E90C6 CC0AC . . .

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In each of the above counting systems, pi’s expanded form retains the usual properties of irrational numbers: the digits don’t ever terminate, nor settle into a repetitive pattern. It also isn’t possible, in any of these counting systems, to express pi as a/b, where a and b are whole numbers in that base. However, in base-pi, the number pi is simply written this way, in its entirety: 10. Also, the square of pi is written 100, pi-cubed is written 1000, etc. However, if you want to try to figure out how to write, say, the decimal number ten, in base-pi, best of luck to you.

# A Number Theory Puzzle Involving Primes, Perfect Numbers, and “Paraperfect” Numbers

There is something about the definitions of prime and perfect numbers that always struck me as rather odd. Prime numbers are those which no factors other than themselves, and one. Perfect numbers, on the other hand, equal the sum of all of their factors, excluding themselves, but including one. The first two examples of perfect numbers are 6 (which equals 1 + 2 + 3) and 28 (which equals 1 + 2 + 4 + 7 + 14). Perfect numbers are far more rare than primes.

The thing I find annoying is the exclusion of one, as a factor, from one of these definitions, but not the other. I therefore decided to give a name to a new type of number:  one which equals the sum of its factors, excluding itself AND the number one. The first name I thought of, “exceptional numbers,” turns out to have already been taken, so I thought of another, and called these numbers “paraperfect numbers” instead.

Having done that, it was time to start searching for them. I have a reasonably fast mental calculator, but it didn’t take long to figure out that I wasn’t up to this task, so I wrote this program to search for paraperfect numbers:

It’s written in BASIC, an archaic computer language I learned in high school, and, as you can see, I am a horribly sloppy programmer. A better programmer would have written a program for the same purpose, but with only about half this length. Be that as it may, though, the program does work. I’ve had it running for a few minutes now.

It’s gotten past 22,000 — and has found no paraperfect numbers at all. This is not what I expected . . . and now I am wondering if any exist. Right now, of course, “no paraperfect numbers exist” is a mere conjecture. If I can prove it, it will be a theorem. However, I don’t know nearly enough about number theory to write such a proof.

I could use some help. If anyone does find a paraperfect number, please leave a comment on this post identifying your find. If anyone can prove — or simply explain to me — why there are no paraperfect numbers, if that is the case, please let me know that as well.

The program is still running, and has now passed 25,000 without a paraperfect-number find. I guess I’ll leave it running for a while. Any help with this puzzle would be much appreciated.

[Later:  see the comments for the rest of the story on these elusive numbers.]