Numbers
Continuing the porting of stuff from betaveros.stash, and adding more stuff.
Mnemonic
Here's my mnemonic table for digits, inspired by an old Martin Gardner column. I wrote from memory the first 132 digits of 2012! correctly at IMO 2012 with this table. I had remembered more, but unfortunately, if I recall correctly, I confused myself over whether I had encoded a 5 or a 2 by the S of "nose", because this is supposed to be more of a phonetic code than a spelling one — otherwise double letters would be confusing and lots of randomly appearing digraphs would be wasted, because English is weird.
Although most of the associations are completely changed, the set of unused consonants is the same. They spell WHY.
| digit | consonant(s) | mnemonic |
|---|
| 1 | L | l33t |
| 2 | Z(, J, CH, TH) | l33t |
| 3 | M | tilt head sideways |
| 4 | F(, V) | Four |
| 5 | S(, SH) | l33t |
| 6 | G(, K) | l33t |
| 7 | T(, D) | l33t |
| 8 | B(, P) | l33t |
| 9 | N | Nine |
| 0 | R | zeRo, or it's a Round circle |
Bits & Primes
2095133040 has 1600 factors, the most of any positive integer under 231 − 1. Ref: A002182: highly composite numbers, def. 1
There are 105097565 (1.05e8) primes under 231 − 1.
Miller-Rabin primality testing does not miss any composites:
- below 231 if the first 3 primes (2, 3, 5) are used as witnesses.
- below 232 if the first 4 primes (2, 3, 5, 7) are used as witnesses.
- below 264 if the first 12 primes (2 to 37 inclusive) are used as witnesses.
See A014233.
To verify implementations: there are 82025 primes beneath (1 << 20) and 37871 primes between (1 << 40) and (1 << 40) + (1 << 20).
Primes in Decimal
- 1881881
- n − 1 factorization: 23 × 5 × 7 × 11 × 13 × 47
- smallest primitive root: 6
- 99990001
- n − 1 factorization: 24 × 32 × 54 × 11 × 101
- smallest primitive root: 13
- 140000041
- n − 1 factorization: 23 × 32 × 5 × 157 × 2477
- smallest primitive root: 28
- 987654323
- n − 1 factorization: 2 × 701 × 704461
- smallest primitive root: 2
- 999299999
- n − 1 factorization: 2 × 23 × 21723913
- smallest primitive root: 11
- 999992999
- n − 1 factorization: 2 × 499996499
- smallest primitive root: 11
- 999999001
- n − 1 factorization: 23 × 33 × 53 × 7 × 11 × 13 × 37
- smallest primitive root: 17
- 999999929
- n − 1 factorization: 23 × 124999991
- smallest primitive root: 3
- 1000000007
- n − 1 factorization: 2 × 500000003
- smallest primitive root: 5
- 1000000009
- n − 1 factorization: 23 × 32 × 7 × 1092 × 167
- smallest primitive root: 13
- 1000000021
- n − 1 factorization: 22 × 3 × 5 × 19 × 739 × 1187
- smallest primitive root: 2
- 1234567891
- n − 1 factorization: 2 × 32 × 5 × 3607 × 3803
- smallest primitive root: 3
- 2000000011
- n − 1 factorization: 2 × 3 × 5 × 66666667
- smallest primitive root: 2
Primes in Hexadecimal
0xdefaced
- n − 1 factorization: 22 × 32 × 5 × 1298951
- smallest primitive root: 6
0xfacade5
- n − 1 factorization: 22 × 3 × 21914579
- smallest primitive root: 6
0x37beefed
- n − 1 factorization: 22 × 3 × 5 × 1373 × 11353
- smallest primitive root: 6
0x3c0ffee1
- n − 1 factorization: 25 × 7 × 2113 × 2129
- smallest primitive root: 3
0x3de1f1ed
- n − 1 factorization: 22 × 11 × 23595857
- smallest primitive root: 3
0x3efface5
- n − 1 factorization: 22 × 32 × 37 × 47 × 16883
- smallest primitive root: 2
0x5eedbed5
- n − 1 factorization: 22 × 7 × 53 × 199 × 5393
- smallest primitive root: 10
Floating Point
- Single-precision: sign 1b, exponent 8b, fraction 23+1b implied (= 6 ~ 9 decimal sigfigs)
- Double-precision: sign 1b, exponent 11b, fraction 52+1b implied (= 15 ~ 17 decimal sigfigs)
Special cases:
- Exponent = 0
- fraction = 0: (±) zero
- fraction ≠ 0: “subnormal” number with implied bit set to 0 instead
- Exponent = (FF or 3FF, maximum value in allocated bits)
- fraction = 0: (±) infinity
- fraction ≠ 0: NaN (sign ignored)
- top explicit fraction bit = 1: “quiet NaN”
- top explicit fraction bit = 0 (and rest ≠ 0): “signaling NaN”