nbsp; Note: 20110218 - See the following distributed application case study that uses Collatz to research concurrency issues.
http://wiki.eclipse.org/EclipseLink/Examples/Distributed
I ran into hailstone numbers again recently when i was writing a very quick benchmark and volumetric test for JVMs deployed on my server farm recently. I needed a way to test mixed integer and floating point performance using the unbounded BigInteger java class. what better way than writing a small (3n +1) algorithm - like i did in my youth in 1984 after i read Brian Hayes' Scientific american column.
it turns out that you get much better performace by using the following java virtual machines in descending order. SUN 1.6 server 64bit, Oracle JRockit 32/64bit, the latest 1.6.0_23 SUN and lastly older SUN 1.6 JVM's .
See: http://en.m.wikipedia.org/wiki/Collatz_conjecture?wasRedirected=true
The scientific part of this exploration would be attempt to verify the Collatz theorem past 100 billion - which would normally result in a Long.MAX_VALUE overflow past 2^63 of the max height reached. Ideally i would like to find a hainstone number that stays above the 4-2-1 squence indefinitely - this will take a lot of computer power.
The good news is - we now have access to a lot of computing power (gone are the days of running 3n+1, game of life and mandelbrot work over night on my TRS-80 - we have multicore and RMI for scientific distributed computing. What we really need is time on a supercomputing architecture or distrubution of the BigInteger ranges on a massive number of distributed processors in parallel.
After 1 week of BigInteger iteration on around 20 JVM's calculating a half trillion sequences - I am currently at the following maximum path.
For n = 404,970,804,222 we reach a path of 1308 and a max of 2,662,567,439,048,656
Problem:
Every number under the function where odd numbers increase to 3n + 1 and even numbers are reduced by n/2 eventually reach 1 and loop forever in the 4-2-1 sequence.
Optimizations:
1 - Mike Roosendaal: all odd number transformations are always followed by an even transformation - therefore both steps can be done together.
If odd : n = (3n + 1)/2 = (3/2)n + 1/2 = n + n/2 + 1/2
or using shifts (my optimization) : (n >> 1) + n + 1
Notice that the transformation for odd
[ (n << 1) + n + 1 ] is nearly identical to the double transformation for odd with an immediate even (n >> 2) - as in
[ (n >> 1) + n + 1 ] except for the reversal of a shift left to a shift right.
Architecture:
We can narrow down our choice of architecture to two opposite examples. Either we have multiple clients that asynchronously ask a central server for units of work (the
SETI@Home model), or we run multiple servers controlled from a single client.
Option 1: @ManyToOne
Option 2: @OneToMany
Hardware:
1 - SunFire Virtual Linux (dual core VM)
1 - Core i7-920 (2.7GHz 4-core/8-thread)
4 - P4-630 (3.0 GHz 1-core/2-thread)
4 - P4-531 (3.0 GHz 1-core/2-thread)
2 - E8400 (3.0 GHz - 2-core/2-thread)
1 - Q6600 (2.4 GHz - 4-core/4-thread)
1 - T4400 (2.2 GHz - 2-core/2-thread)
1 - P4- (2.0 GHz - 1-core/1-thread)
1 - P4 - (1.6 GHz - 1-core/1-thread)
2 - XMOS XS-1/G4 - (400 MHz - 4-core/32-thread)
80 - P8X-32 (20 MHz - 8-core/8-thread)
Software:
For ease of use i am running on various versions of the Java Virtual Machine (which is roughly performant with compiled native C for integer computation) on all Windows and Linux boxes.
For the XMOS chips i run XC (parallel C) and the Parallax chips run SPIN mixed with machine language.
SUN Hotspot JVM 1.6.0
Oracle JRockit JVM 1.6.0
Java client before extreme optimization (object pooling, loop truncation...) and distribution:
public List hailstoneSequence(BigInteger start) {
List sequence = new ArrayList();
if(start.equals(BigInteger.ZERO) || start.equals(BigInteger.ONE)) {
return sequence;
}
BigInteger current = start;
while (!current.equals(BigInteger.ONE)) {
if(current.testBit(0)) { // test odd
current = current.shiftLeft(1).add(current).add(BigInteger.ONE);
} else {
current = current.shiftRight(1);//.divideAndRemainder(TWO)[0];
}
sequence.add(current);
}
return sequence;
}
public static void main(String[] args) {
// get parameters
int sector = Integer.parseInt(args[0]);
int mult = Integer.parseInt(args[1]);
BenchMark aBench = new BenchMark();
StringBuffer buffer = null;
long x = 0;
//long lastVal = Long.MAX_VALUE;
long lastVal = 4294967296L;
long startTime = GregorianCalendar.getInstance().getTimeInMillis();
for(int y=0;y<1;y++) {
startTime = GregorianCalendar.getInstance().getTimeInMillis();
for(x=0;x< lastVal;x++) {
x = x + 1 - 1;
//System.out.println(buffer.toString());
}
long endTime = GregorianCalendar.getInstance().getTimeInMillis() - startTime;
System.out.println(y + ":" + x + "," + endTime + "," + (x/((1 + endTime)/1000)) + " it/sec");
}
long maxPath = 0; // path should fit in 64 bits
BigInteger maxValue = BigInteger.ONE;
startTime = GregorianCalendar.getInstance().getTimeInMillis();
BigInteger currentMax = BigInteger.ONE;
boolean milestone = false;
String prefix = null;
long rangeStart = (sector * mult) * 1048576L;//67108864L;//4294967296L;
long rangeEnd = rangeStart + (mult * 1048576L);//67108864L);//4294967296L);
long rangeInterval = (mult * 65536);//1048576;//67108864L;//134217728L;//268435456L;
System.out.println("Proc cores: " + Runtime.getRuntime().availableProcessors());
System.out.println("Runtime : " + Runtime.getRuntime());
System.out.println("Short max: " + Short.MAX_VALUE);
System.out.println("Integer max: " + Integer.MAX_VALUE);
System.out.println("Long max: " + Long.MAX_VALUE);
System.out.println("Partition: " + rangeInterval);
System.out.println("Range: " + rangeStart + " to: " + rangeEnd);
long currentNumber = rangeStart;
List list = null;
/**
* We use a double loop to show interim progress by splitting up the search space into quadrants
*/
for(long interval = 0;interval < 16;interval++) { // sectors are dived by two (we only test odd numbers)
currentNumber = currentNumber + 1;
for(long index=1;index maxPath) {
milestone = true;
maxPath = list.size();
prefix = "P";
}
if(currentMax.compareTo(maxValue) > 0) {
if(milestone) {
prefix = "PM";
} else {
milestone = true;
prefix = "M";
}
maxValue = currentMax;
}
if(milestone) {
buffer = new StringBuffer(prefix);
buffer.append(",N,");
buffer.append(interval);
buffer.append(",");
buffer.append(index);
buffer.append(",\t");
buffer.append(currentNumber);
buffer.append(",L,");
buffer.append(list.size());
buffer.append("\t");
buffer.append(",M,");
buffer.append(currentMax); // BigInteger implements Comparable
//buffer.append(list.toString());
buffer.append("\t");
buffer.append(",T,");
buffer.append(GregorianCalendar.getInstance().getTimeInMillis() - startTime);
buffer.append("\t");
if((currentMax.compareTo(BigInteger.valueOf(Long.MAX_VALUE)) > 0)) {
buffer.append("2^63+,");
buffer.append(currentMax.subtract(BigInteger.valueOf(Long.MAX_VALUE)));
buffer.append(",");
} else {
buffer.append(",,");
}
buffer.append(",F,");
buffer.append(Runtime.getRuntime().freeMemory());
System.out.println(buffer.toString());
milestone = false;
}
}
// increment
currentNumber += 2;
}
}
Power:
Todo: I need to get some single crystal PV boards on my roof and join microFIT so i can get $0.80 to offset my 2 KiloWatts (Note to OPA - my usage ranges from .5KWh to 3KWh depending on what simulations are running - i am not a GO).
So far in the basement i am running about 14-18 A at 120V AC =
about 1.7 to 2 KWh or about $0.50/hour when i have 10-14 boxes running.
It may be more efficient and cheaper to replicate my ECC cloud instance that is running about $
0.14 /h (as long as i remember to stop and detach all the instances)
C++
// collatz_vs10.cpp : Defines the entry point for the console application.
#include "stdafx.h"
__int64 hailstoneMax(__int64 start) {
__int64 maxNumber = 0;
__int64 num = start;
__int64 path = 0;
while(num > 4) {
//printf("%I64d ", maxNumber);
if((num % 2) > 0) {
num = (num << 1) + num + 1; // odd
} else {
num >>= 1; // even
}
if(num > maxNumber) {
maxNumber = num;
}
path++;
}
return maxNumber;
}
int _tmain(int argc, _TCHAR* argv[]) {
__int64 num = 27;
//unsigned long long num = 1;
__int64 maxNumber = 0;
__int64 newMax = 0;
unsigned long long path = 0;
unsigned long long maxPath = 0;
__int64 MAX = (1 << 40); // dont use long long
while(num < MAX) {
newMax = hailstoneMax(num);
if(newMax > maxNumber) {
printf("\n%I64d,\t%I64d",num, newMax);
maxNumber = newMax;
//printf("\n%d,\t%I64d",num,maxNumber); // or I64u, %llu (do not work properly)
}
num += 2;
}
return 0;
}
Alternate Platforms
Just as an aside - I wrote some code for the current leaders in multicore microcontrollers - XMOS and Parallax.
The 4-core/32-thread XMOS G4 is easier to develop for because it runs compiled XC (C with parallel and concurrency language additions) instead of SPIN bytecode like the 8-core/8-thread Parallax Propeller (It can run assembly) - we can also use a standard Eclipse.org Eclipse IDE.
Here is some
XMOS parallel XC code for 32 bit integers.
#include
#include //http://www.xmos.com/discuss/viewtopic.php?f=6&t=255
#define PERIOD 20000000
// Processor 0 = 4 green + 3r/g LED + 16 I/O, 4 push-buttons
//out port cledB0 = PORT_BUTTONLED;//PORT_CLOCKLED_0;
out port cled0 = PORT_BUTTONLED;//PORT_CLOCKLED_0;
// anode 4 bit ports
//out port cled0 = PORT_CLOCKLED_0;
// Processor 1 = 3r/g LED + 16 I/O
out port cled1 = PORT_CLOCKLED_1;
// Processor 1 = 3r/g LED + 16 I/O
out port cled2 = PORT_CLOCKLED_2;
// Processor 1 = 3r/g LED + 32 I/O
out port cled3 = PORT_CLOCKLED_3;
// cathode 1 bit ports
out port cledG = PORT_CLOCKLED_SELG;
out port cledR = PORT_CLOCKLED_SELR;
/**
* http://en.wikipedia.org/wiki/XSwitch#XS1-G4_Switch
*/
unsigned long number = 27;
unsigned long maximum = 1 << 18;//138367; // 32 bit registers top out at a 4 billion max for 138367
// Compute the hailstone maximum
unsigned long hailstoneMax(unsigned long start) {
unsigned long maxNumber = 0;
unsigned long number = start;
while(number > 1) {
if((number % 2) > 0) {
number = (number << 1) + number + 1; // odd
} else {
number = number >> 1; // even
}
if(number > maxNumber) {
maxNumber = number;
}
}
return maxNumber;
}
void hailstoneSearch(int coreId, out port led, unsigned long start, unsigned long end) {
unsigned long number = 27;
unsigned long maxNumber = 0;
unsigned long newMax = 0;
int flip = 0;
//write_pswitch_reg(get_core_id(), XS1_PSWITCH_PLL_CLK_DIVIDER_NUM, 0x80);
while(number < end) {
newMax = hailstoneMax(number);
if(newMax > maxNumber) {
maxNumber = newMax;
// TODO: send message to other cores
// UART printing really slows down the cores
/*if(coreId < 1) { // only core 0 prints
printuint(number);
printchar(',');
printchar('\t');
printuintln(maxNumber);
}*/
if(flip > 0) {
flip = 0;
led <: 0b1111;
} else {
flip = 1;
led <: 0b0000;
}
}
number = number + 2;
}
printint(coreId); // print core id when finished
}
// Search a range of integers for their hailstone maximums
void hailstoneSearch0(int coreId, out port led, out port redCathode, out port greenCathode,
unsigned long start, unsigned long end) {
redCathode <: 0b1111;
//cledB0 <: 0b1111;
//greenCathode <: 0b1111;
printuint(start);
printchar('-');
printuintln(end);
hailstoneSearch(coreId, led, start, end);
// reduce temperature by lowering the PLL multiplier
write_pswitch_reg(get_core_id(), XS1_PSWITCH_PLL_CLK_DIVIDER_NUM, 0x80);
while(1);
}
int main() {
// concurrent threads p.33 http://www.xmos.com//system/files/xcuser_en.pdf
par {
on stdcore [0]: hailstoneSearch0(0, cled0,cledR,cledG,number, maximum);
on stdcore [1]: hailstoneSearch(1, cled1,number, maximum);
on stdcore [2]: hailstoneSearch(2, cled2,number, maximum);
on stdcore [3]: hailstoneSearch(3, cled3,number, maximum);
}
return 0;
}
For the
Propeller I wrote a very quick SPIN program that only runs in 1 of the 8 cogs on a parallax propeller microcontroller. This code needs to be parallized and also converted to assembly along with a need for a fixed point unlimited length integer library object. It currently cannot compute sequences past 113381 as this will overload the built in 32 bit register length.
First Assembly language (this is my first routing in Propeller assembly - it is a testament to the tutorial by deSilva from 2007 - I was able to write this function after 2 hours and 14 pages of the
PDF.
DAT
org 0
entry
RDLONG _nextVal, PAR ' read from shared ram (7-22 cycles)
:iterate
ADD _path, #1 ' increment path
MOV _bit0, #1 ' create mask
AND _bit0, _nextVal WZ ' check bit 0 - affect zero flag
IF_NE JMP #:mul3x1
:div2 ' if even we divide by 2
SHR _nextVal, #1 ' divide by 2
MOV _bit0, _nextVal ' check for 1 value == finished
CMP _nextVal, #1 WZ
IF_E JMP #:finish
JMP #:iterate ' return to top of loop
:mul3x1 ' if odd we transform by 3n + 1
MOV _3rdVal, _nextVal
SHL _nextVal, #1 ' multiply by 2
ADD _nextVal, _3rdVal ' add to multiply by 3
ADD _nextVal, #1 ' add 1
:maxValue ' check for maximum value
MIN _maxVal, _nextVal ' VERY ODD (max is actually min)
JMP #:iterate ' return to top of loop
:finish
SUB _path, #1 ' we discount the first path count
'MOV _nextVal, _path ' copy path to return val
MOV _nextVal, _maxVal ' copy maxVal to return value
WRLONG _nextVal, PAR ' write back to hub ram (thank you deSilva for reverse flow explanation)
' WRLONG _path, PAR
:endlessLoop
JMP #:endlessLoop ' keep the cog running
_3rdVal long $00000000
_nextVal long $00000000 '
_maxVal long $00000000
_path long $00000000
_bit0 long $00000000
FIT 496 ' deSilva (16 I/O registers in 496-511
Spin (interpreted bytecode)
CON
_clkmode = xtal1 + pll16x
_xinfreq = 5_000_000
VAR
' shared display RAM for the 4 display cogs
long buffer[32]
long Stack0[64] ' Stack Space
byte Cog[7] ' Cog ID
long randomNum
OBJ
SER : "Parallax Serial Terminal.spin"
STR :"STREngine.spin"
PUB main | milestone,start,number,index, lRec,x,i, mIndex, mValue, path,height, maxPath, maxHeight
' wait for user to switch to terminal
waitcnt((clkfreq / 1_000 * 4) + cnt)
maxPath := 0
maxHeight := 0
milestone := 0 ' track whether we got a path or max height hit
ser.Start(115_200)'31,30,0,38400)
ser.Home
ser.Clear
ser.Str(string("Collatz Conjecture", ser#NL))
'Cog[0] := cognew(Push8seg(0,buffer,2,3,4, 24_000,0), @Stack0) + 1
' main loop
repeat x from 1 to 113381 step 2
start := x'77031'27
path := 0
height := 0
number := start
repeat until number == 1
' if odd transform by 3n+1, else n/2
if (number // 2) == 0
number := number >> 1
else
number := (number << 1) + number + 1
path := path + 1
if height < number
height := number
' check maximums
if maxHeight < height
maxHeight := height
milestone := 1 ' flag a hit
if maxPath < path
ser.Str(string(ser#NL))
maxPath := path
if milestone > 0
ser.Str(string("PM: "))
else
ser.Str(string(" P: "))
milestone := 1 ' flag a hit
else
if milestone > 0
ser.Str(string(ser#NL))
ser.Str(string(" M: "))
' print out result if a new record
if milestone > 0
ser.Str(STR.numberToDecimal(x,8))
ser.Str(string(" Path: "))
ser.Str(STR.numberToDecimal(path,8))
ser.Str(string(" Max: "))
ser.Str(STR.numberToDecimal(height,31))
milestone := 0
' wait to allow the port to catch up before closing
waitcnt((clkfreq / 1_000 * 4) + cnt)
ser.stop
Here, also is a 1992 version of the hailstone number generator in Smalltalk
Results: Full results for the first 360 billion integers are pending at the end of this week, but here are some interim maximum values for the hailstone sequence.
Short max: 32767
Integer max: 2147483647
Long max: 9223372036854775807
Partition: 1073741824
type, path, max,
PM, 2, 2
PM, 3, 7 ,16 ,T,1 ,,,
PM, 7, 16 ,52 ,T,1 ,,,
P, 9, 19 ,52 ,T,1 ,,,
M, 15, 17 ,160 ,T,2 ,,,
P, 19, 20 ,88 ,T,2 ,,,
P, 25, 23 ,88 ,T,2 ,,,
PM, 27, 111 ,9232 ,T,2 ,,,
P, 55, 112 ,9232 ,T,4 ,,,
P, 73, 115 ,9232 ,T,5 ,,,
P, 97, 118 ,9232 ,T,6 ,,,
P, 129, 121 ,9232 ,T,9 ,,,
P, 171, 124 ,9232 ,T,12 ,,,
P, 231, 127 ,9232 ,T,16 ,,,
M, 255, 47 ,13120 ,T,17 ,,,
P, 313, 130 ,9232 ,T,23 ,,,
P, 327, 143 ,9232 ,T,25 ,,,
M, 447, 97 ,39364 ,T,51 ,,,
M, 639, 131 ,41524 ,T,69 ,,,
P, 649, 144 ,9232 ,T,69 ,,,
PM, 703, 170 ,250504 ,T,69 ,,,
P, 871, 178 ,190996 ,T,70 ,,,
P, 1161, 181 ,190996 ,T,72 ,,,
M, 1819, 161 ,1276936 ,T,75 ,,,
P, 2223, 182 ,250504 ,T,79 ,,,
P, 2463, 208 ,250504 ,T,80 ,,,
P, 2919, 216 ,250504 ,T,83 ,,,
P, 3711, 237 ,481624 ,T,87 ,,,
M, 4255, 201 ,6810136 ,T,91 ,,,
M, 4591, 170 ,8153620 ,T,93 ,,,
P, 6171, 261 ,975400 ,T,99 ,,,
M, 9663, 184 ,27114424 ,T,117 ,,,
P, 10971, 267 ,975400 ,T,123 ,,,
P, 13255, 275 ,497176 ,T,133 ,,,
P, 17647, 278 ,11003416 ,T,159 ,,,
M, 20895, 255 ,50143264 ,T,187 ,,,
P, 23529, 281 ,11003416 ,T,241 ,,,
PM, 26623, 307 ,106358020 ,T,254 ,,,
P, 34239, 310 ,18976192 ,T,288 ,,,
P, 35655, 323 ,41163712 ,T,294 ,,,
P, 52527, 339 ,106358020 ,T,376 ,,,
P, 77031, 350 ,21933016 ,T,487 ,,,
P, 106239, 353 ,104674192 ,T,623 ,,,
P, 142587, 374 ,593279152 ,T,811 ,,,
P, 156159, 382 ,41163712 ,T,880 ,,,
P, 216367, 385 ,11843332 ,T,1179 ,,,
P, 230631, 442 ,76778008 ,T,1253 ,,,
P, 410011, 448 ,76778008 ,T,2172 ,,,
P, 511935, 469 ,76778008 ,T,2759 ,,,
P, 626331, 508 ,7222283188 ,T,3384 ,,,
M, 665215, 441 ,52483285312 ,T,3598 ,,,
M, 704511, 242 ,56991483520 ,T,3837 ,,,
P, 837799, 524 ,2974984576 ,T,4582 ,,,
M, 1042431, 439 ,90239155648 ,T,5726 ,,,
P, 1117065, 527 ,2974984576 ,T,6161 ,,,
M, 1212415, 328 ,139646736808 ,T,6701 ,,,
M, 1441407, 367 ,151629574372 ,T,8010 ,,,
P, 1501353, 530 ,90239155648 ,T,8360 ,,,
P, 1723519, 556 ,46571871940 ,T,9639 ,,,
M, 1875711, 370 ,155904349696 ,T,10532 ,,,
M, 1988859, 427 ,156914378224 ,T,11194 ,,,
M, 2643183, 430 ,190459818484 ,T,15091 ,,,
M, 2684647, 399 ,352617812944 ,T,15339 ,,,
M, 3041127, 363 ,622717901620 ,T,17477 ,,,
M, 3873535, 322 ,858555169576 ,T,22541 ,,,
P, 2298025, 559 ,46571871940 ,T,13033 ,,,
P, 3064033, 562 ,46571871940 ,T,17616 ,,,
P, 3542887, 583 ,294475592320 ,T,20526 ,,,
P, 3732423, 596 ,294475592320 ,T,21685 ,,,
M, 4637979, 573 ,1318802294932 ,T,27248 ,,,
P, 5649499, 612 ,1017886660 ,T,33549 ,,,
M, 5656191, 400 ,2412493616608 ,T,33592 ,,,
M, 6416623, 483 ,4799996945368 ,T,38375 ,,,
M, 6631675, 576 ,60342610919632 ,T,39740 ,,,
P, 6649279, 664 ,15208728208 ,T,39857 ,,,
P, 8400511, 685 ,159424614880 ,T,51033 ,,,
P, 11200681, 688 ,159424614880 ,T,69226 ,,,
P, 14934241, 691 ,159424614880 ,T,93871 ,,,
P, 15733191, 704 ,159424614880 ,T,99175 ,,,
M, 19638399, 606 ,306296925203752 ,T,125377 ,,,
P, 31466383, 705 ,159424614880 ,T,206680 ,,,
P, 36791535, 744 ,159424614880 ,T,243861 ,,,
M, 38595583, 483 ,474637698851092 ,T,256522 ,,,
P, 63728127, 949 ,966616035460 ,T,436156 ,,,
M, 80049391, 572 ,2185143829170100 ,T,554782 ,,,
M, 120080895, 438 ,3277901576118580 ,T,850400 ,,,
P, 127456255, 950 ,966616035460 ,T,905504 ,,,
P, 169941673, 953 ,966616035460 ,T,1226118 ,,,
M, 210964383, 475 ,6404797161121264 ,T,1540036 ,,,
P, 226588897, 956 ,966616035460 ,T,1659884 ,,,
P, 268549803, 964 ,966616035460 ,T,1984199 ,,,
M, 319804831, 592 ,1414236446719942480 ,T,2383830 ,,,
P, 537099607, 965 ,966616035460 ,T,4104391 ,,,
P, 670617279, 986 ,966616035460 ,T,5175958 ,,,
P, 1341234558, 987 ,966616035460 ,T,10515479 ,,,
M, 2379584155, 763 ,7125885122794452160 ,T,18815293 ,,,
P, 2384416993, 993 ,966616035460 ,T,18855328 ,,,
P, 2511978395,1006 ,966616035460 ,T,19923962 ,,,
P, 2610744987,1050 ,966616035460 ,T,20751706 ,,,
P, 4578853915,1087 ,966616035460 ,T,37063783 ,,,
P, 4890328815,1131 ,319497287463520 ,T,39763292 ,,,
P, 9780657630,1132 ,319497287463520 ,T,81997346 ,,,
M, 10829712411, 672 ,15781722338690299312 ,T,90904965 2^63+,6558350301835523505,,
M, 11371756681, 729 ,18144594937356598024 ,T,95810690 2^63+,8921222900501822217,,
M, 12987343015, 556 ,20722398914405051728 ,T,109846948 2^63+,11499026877550275921,,
P, 13040876841,1135 ,319497287463520 ,T,110329343 ,,,
P, 13371194527,1210 ,319497287463520 ,T,113344783 ,,,
Range:635655159808 to: 652835028992
P, 639953863463,1319, ,2662567439048656 ,41245333 ,
Range: 51539607552 to: 68719476736
M, 51739336447, 770 ,114639617141613998440 ,T,3437912 2^63+,105416245104759222633,,
Range: 17179869184 to: 34359738368
P,17828259369,1213 ,319497287463520 ,T,6056083 ,,,
Range: 34359738368 to: 51539607552
P,1,223038545, 35656518738,1214 ,319497287463520 ,T,17634743 ,,,
P,12,297384717, 47542024985,1217 ,319497287463520 ,T,125367559 ,,,F,54935512
Range: 51539607552 to: 68719476736
P,11,38599019, 63389366646,1220 ,319497287463520 ,T,113845586 ,,,F,66350984
Range: 68719476736 to: 85899345920
Range: 85899345920 to: 103079215104
Range: 103079215104 to: 120259084288
P,8,1023057667, 112692207371,1226 ,319497287463520 ,T,356397390 ,,,F,4636736
Range: 120259084288 to: 137438953472
P,12,417148475, 133561134663,1234 ,319497287463520 ,T,480482552 ,,,F,4361648
Range: 171798691840 to: 188978561024
Range: 206158430208 to: 223338299392
P,4,606173317, 211059570825,1245 ,319497287463520 ,T,141893594 ,,,F,11744352
P,12,314731323, 219358063431,1289 ,319497287463520 ,T,381152328 ,,,F,15132528
Range: 240518168576 to: 257698037760
Range: 274877906944 to: 292057776128
Range: 292057776128 to: 309237645312
P,10,932908789, 303728103167,1305 ,2662567439048656 ,T,260241794 ,,,F,37314808
Range: 326417514496 to: 343597383680
Range: 395136991232 to: 412316860416
P,9,170136565,404970804222,1308 ,2662567439048656 ,114658773 ,
Range: 635655159808 to: 652835028992
P,4,3736355,639953863463,1319 ,2662567439048656 ,41245333 ,
hold from Current Research: As I was optimizing my algorithm from a brute-force/naive approach to one that takes advantage of properties of the Lothar Collatz (3n+1) sequence I did some research on what the current state of Collatz investigation is current at. Here are some interesting results. Main Wiki site
http://en.wikipedia.org/wiki/Collatz_conjecture Tomás Oliveira e Silva at Universidade de Aveiro has computed the sequence to
http://www.ieeta.pt/~tos/3x+1.html Eric Roosendaal has offered several usefull optimizations http://www.ericr.nl/wondrous/index.html Ken Korrow
http://www-personal.ksu.edu/~kconrow/gentrees.html References: Signed Short MAX is 32767 (2^15 - 1)
Signed Integer MAX is (2 billion) 2,147,483,647 (2^31 - 1)
Signed Long MAX is (9 quintillion) 9,223,372,036,854,775,807 (2 ^63 - 1)
