Prime Intirety

Since ancient times humanity knew that there are infinitely many primes — though countable, writing a complete list of every prime is impossible if one intends to finish.
However, in practice one often only considers a minute subset of the naturals to work with and think about. When writing low-level languages like C, one is nearly forced to forget about almost every natural number — the data type u_int_32, for example, is only capable of representing \{\mathbb{N}_0\ni n<2^{32}\}.
Therefore, it is possible to produce a complete list of every prime representable in thirty-two bits using standard bit pattern interpretation — the entirety of the first 203\,280\,221 primes.

Generating said list took about two minutes on a 4GHz Intel Core i7 using an elementary sieve approach written in C compiled with gcc -O2.
All primes are stored in little-endian format and packed densely together, requiring four bytes each.

Using the resulting file, one can quickly index the primes, for example p_{10^7} = 179\,424\,691 = \text{ab1cdb3}_{16} (using zero-based indexing). Since each prime is stored using four bytes, the prime’s index is scaled by a factor of four, resulting in its byte index.

dd status=none ibs=1 count=4 if=primes.bin skip=40000000 | xxd 
00000000: b3cd b10a                                ....

Source code: intirety.c
Prime list: primes.bin (775.5 MiB)

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Truth

Proposition calculus deals with statements and the relation between statements, where each of them can only be in one of two states; \vdash p \lor \lnot p. Therefore, when working with finitely many connected propositions, one can algorithmically determine all possible truth values of all atomic and thus connected propositions.

Truth is command-line tool which was written to precisely perform those computations; computing a logical expression’s truth value. Download link: truth.py
A list of all supportet operators can be seen by invoking the tool with a --help flag.
This project was inspired by Albert Menne’s Einführung in die Logik1; the operator syntax used is similar to his, translated to be 7-bit-ASCII-compatible.

Truth can be used to either verify universally true statements, e.g. tertium non datur and a property of the replication, verum sequitur ex quodlibet.

-(p&-p) <-> 1  ,  1 <- p
1 0010   1  1     1 1  0
1 1001   1  1     1 1  1

Though not only absolute truth, but also complete relational equivalence between two expressions can be shown.

(p->q)|(r>-<s) <-> q|(r|s)&-(r&s)|-p
 01 0 1 0 0 0   1  00 000 01 000 110
 10 0 0 0 0 0   1  00 000 01 000 001
 01 1 1 0 0 0   1  11 000 01 000 110
 11 1 1 0 0 0   1  11 000 01 000 101
 01 0 1 1 1 0   1  01 110 11 100 110
 10 0 1 1 1 0   1  01 110 11 100 101
 01 1 1 1 1 0   1  11 110 11 100 110
 11 1 1 1 1 0   1  11 110 11 100 101
 01 0 1 0 1 1   1  01 011 11 001 110
 10 0 1 0 1 1   1  01 011 11 001 101
 01 1 1 0 1 1   1  11 011 11 001 110
 11 1 1 0 1 1   1  11 011 11 001 101
 01 0 1 1 0 1   1  00 111 00 111 110
 10 0 0 1 0 1   1  00 111 00 111 001
 01 1 1 1 0 1   1  11 111 00 111 110
 11 1 1 1 0 1   1  11 111 00 111 101

Complete contravalence can also be shown.

-(p/-p>-<0)|p->q<-r >-< p&-q&r
0 0110 1 0 101 01 0  1  001000
0 1101 1 0 110 01 0  1  111000
0 0110 1 0 101 11 0  1  000100
0 1101 1 0 111 11 0  1  100100
0 0110 1 0 101 01 1  1  001001
0 1101 1 0 010 00 1  1  111011
0 0110 1 0 101 11 1  1  000101
0 1101 1 0 111 11 1  1  100101

1Menne, Albert: Einführung in die Logik. Bern: Franke, 1966. (= Dalp-Taschenbücher; 384 D)