## Winter MMXVIII

I
,O;
main(
){char*
Q,_[]={73
,44,79,59,2
,109,97,105,2
,110,40,41,123,
99,104,97,114,42,
81,44,95,91,93,61,2
,123,1,48,125,44,42,2
,74,59,102,111,114,40,2
,81,61,95,44,73,61,79,61,
48,59,42,81,59,43,43,81,41,
123,105,102,40,42,81,60,50,41
,102,111,114,40,74,61,95,59,42,
74,59,74,43,43,41,73,60,49,38,38,
112,114,105,110,116,102,40,34,37,42
,99,34,44,52,50,45,79,47,50,45,49,44,
51,50,41,44,73,43,61,112,114,105,110,2,
116,102,40,34,37,100,34,44,42,74,41,44,73
,62,79,63,73,61,48,44,79,43,61,50,44,112,2,
117,116,115,40,34,34,41,58,48,44,73,43,43,60,
49,63,112,114,105,110,116,102,40,34,37,42,99,34
,44,52,50,45,79,47,50,44,52,52,41,58,112,117,116,
99,104,97,114,40,52,52,41,44,73,62,79,63,73,61,48,2
,44,79,43,61,50,44,112,117,116,115,40,34,34,41,58,48,
59,105,102,40,42,81,62,50,41,73,43,43,60,49,63,112,114,
105,110,116,102,40,34,37,42,99,34,44,52,50,45,79,47,50,44
,42,81,41,58,112,117,116,99,104,97,114,40,42,81,41,44,73,62
,79,63,73,61,48,44,79,43,61,50,44,112,117,116,115,40,34,34,41
,58,48,59,125,102,111,114,40,73,61,48,59,73,43,43,60,49,55,59,2
,112,117,116,99,104,97,114,40,52,55,41,41,59,102,111,114,40,73,61
,112,117,116,115,40,34,34,41,59,73,43,43,60,53,59,112,117,116,115,2
,40,34,34,41,41,123,112,114,105,110,116,102,40,34,37,42,99,34,44,51,2
,50,44,52,55,41,59,102,111,114,40,74,61,48,59,74,43,43,60,50,48,59,41,2
,2,112,117,2,116,99,2,2,104,2,97,2,114,2,40,52,2,55,41,2,59,125,2,2,125,2
,2,0},*J;for(Q=_,I=O=0;*Q;++Q){if(*Q<2)for(J=_;*J;J++)I<1&&printf("%*c",42-
O/2-1,32),I+=printf("%d",*J),I>O?I=0,O+=2,puts(""):0,I++<1?printf("%*c",42-O/
2,44):putchar(44),I>O?I=0,O+=2,puts(""):0;if(*Q>2)I++<1?printf("%*c",42-O/2,*Q)
:putchar(*Q),I>O?I=0,O+=2,puts(""):0;}for(I=0;I++<17;putchar(47));for(I=puts("");
I++<5;puts("")){printf("%*c",32,47);for(J=0;J++<20;)putchar(47);}}/////////////////
/////////////////////
/////////////////////
/////////////////////
/////////////////////

Try it online.

## Symbolic Closed-Form Fibonacci

Let $V := \{(a_j)_{j\in\mathbb{N}}\subset\mathbb{C}|a_n=a_{n-1}+a_{n-2}\forall n>1\}$ be the two-dimensional complex vector space of sequences adhering to the Fibonacci recurrence relation with basis $B := ((0,1,\dots), (1,0,\dots))$.
Let furthermore $f: V\to V, (a_j)_{j\in\mathbb{N}}\mapsto(a_{j+1})_{j\in\mathbb{N}}$ be the sequence shift endomorphism represented by the transformation matrix

$A := M^B_B(f) = \begin{pmatrix}1&1\\1&0\end{pmatrix}$.

By iteratively applying the sequence shift a closed-form solution for the standard Fibonacci sequence follows.

## 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)