Introduction:
This program aims to empirically check Pollock's conjecture[1] for 
integers smaller than 2^64.

Briefly, Pollock's conjecture states that every integer can be represented
as the sum of no more than 5 tetrahedral numbers. Where a tetrahedral
number is defined as T(m) = (m^3 - m) / 6

Compiling:
gcc -O3 -o pyramidal pyramidal.c
Headers may have to be changed if using MSVC since I don't believe
Microsoft supports stdint.h or the C99 uintxx_t types but it's been a while
since I checked this.

Usage:
Run pyramidal with a single integer parameter. The parameter indicates which
billion integer block should be examined. For example

user@host:~$ ./pyramidal 1
N1 number in segment 1: 1816
N2 number in segment 1: 1451433
N3 number in segment 1 part 0: 446186613
N4ish coverage of the next segment 1000000000

Output:
The output of this program is statistics on the distribution of integers in
a billion integer slice. In the above output N1 number indicates the number
of tetrahedral numbers in the interval [1, 1,000,000,000] while N2 indicates
the number of integers in that interval representable as the sum of exactly
two tetrahedral numbers. N3 similarly represents the number of integers
requiring axactly three tetrahedral numbers to represent them. For
parameters greater than 1 this line also gives an indication of progress.
N4ish coverege indicates the number of integers in the following segment
- [1,000,000,001, 2,000,000,000] in this case - which can be represented as
the sum of N1, N2 and N3 elements in the given segment plus tetrahedral
numbers in the inteval [1, 10^9]. An output of 1000000000 here indicates
complete coverage. The reason this is referred to as N4ish coverage is that
this is actually proves a stronger statement than whether these numbers can
be represented as the sum of ANY four tetrahedral numbers. If this number
is less than a billion it is still possible that all the integers are
represented and a more complete check should be undertaken. Thusfar coverage
appears complete up to 401 billion.

Background:
In the first published numerical attack on Pollock's conjecture Salzer[2]
enumerated the 241 integers known to require exactly five tetrahedral
numbers, the largest of which is 343,867. 
Afterwards this problem was revisited around 1992 by several mathematicians
who were concerned with checking the conjecture for integers up to 10^9.[3]
The most recent published effort managed to verify the conjecture for
integers up to 40 billion.[4] Using this program along with some help from 
friends and EC2 instance I was able to generate statistics for numbers up to
400 billion, writeup to follow soon on my blog[5].

[1] http://mathworld.wolfram.com/PollocksConjecture.html
[2] Salzer, H. E. and Levine, N. "Table of Integers Not Exceeding 1000000 that are Not Expressible as the Sum of Four Tetrahedral Numbers." Math. Comput. 12, 141-144, 1958.
[3] Waring's Problem for Pyramidal Numbers http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.15.3230&rep=rep1&type=pdf
[4] Decomposing 40 billion integers by four tetrahedral numbers. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.7839&rep=rep1&type=pdf
[5] http://euler.rouxheyns.com