Pseudo Diophantine equation  sum of 3 cubes
Description
J E Patterson 20190928
Introduction
Published 3 cubes solutions can be derived from just a few least significant digits of, what can sometimes be extremely large, numbers. This 37 step program calculates the sum of cubes for 3 numbers using just the 3 least significant digits in each term.
Numbers are truncated to three digits using FRAC. Negative numbers are complimented by having 1000 added to them to make a new positive number. Each 3 digit number is cubed and they are summed. Finally the result is truncated to 3 digits again, using FRAC, leaving just the published 3 cubes sum.
This little discovery is probably obvious to mathematicians, but I found it interesting.
It was also interesting to not truncate the final sum. Using two digits from the first example below the result is 50742. Using three digits the result was 387848042. Using four digits the result was 668216100042. At this point the significant digit range of the hp15c is exceeded so this was calculated using Thomas Okken's Free42 simulator.
Another option is to use Wolfram Alpha. For example 75145817515^{3} + (100000000000  38812075974)^{3} + 23297335631^{3} = 666070721166961397611075000000042. The number of zeros seen increases to just 15 when all the significant digits are in place, producing 529784370469658641488703464414602800000000000000042. Removing the now fully expanded 100000000000000000 leaves a result of 42.
Taking simple cubes produced sums ending in 058 for two digits, 9958 for three, 0042 for four and 99958 for five digits.
Links
The links below are representative of many on this subject. The first is a research paper by Andrew Booker for the number 33. The second is a general article about the solution for 42 which has better researched content than most.
Cracking the problem with 33
The Sum of Three Cubes Problem For 42 Has Just Been Solved
Calculation
Enter the last 3 digits (or more) of each number.
The 3 numbers are now in x, y and z on the stack.
GSB A
The sum of cubes is displayed.
This program produces the same results as a full summation of cubes for sums less than 1000.
Examples
80435758145817515^{3}−80538738812075974^{3}+12602123297335631^{3} = 42
7515 ENTER
5974 CHS ENTER
5631 GSB A
42 is displayed
2^{3}+2^{3}1^{3} = 15
2 ENTER
2 ENTER
1 CHS GSB A
15 is displayed
569936821221962380720^{3}−569936821113563493509^{3}−472715493453327032^{3} = 3
720 ENTER
493509 CHS ENTER
7032 CHS GSB A
3 is displayed
8866128975287528^{3}−8778405442862239^{3}−2736111468807040^{3} = 33
7528 ENTER
62239 CHS ENTER
807040 CHS GSB A
33 is displayed
8866128975287528^{3}−8778405442862239^{3}−2736111468807040^{3} = 33
528 ENTER
862239 CHS ENTER
7040 CHS GSB A
33 is displayed
Program Resources
Labels
Name 
Description 

A 
Calculate cube sum 

B 
Isolate last 3 digits 

C 
Test if negative 

D 
Add 1000 

Storage Registers
Name 
Description 

1 
x1 

2 
x2 

3 
x3 

Program
Line 
Display 
Key Sequence 

Line 
Display 
Key Sequence 

000 



019 
3 
3 

001 
42,21,11 
f LBL A 

020 
10 
÷ 

002 
44 1 
STO 1 

021 
42 44 
f FRAC 

003 
33 
R⬇ 

022 
26 
EEX 

004 
44 2 
STO 2 

023 
3 
3 

005 
33 
R⬇ 

024 
20 
× 

006 
44 3 
STO 3 

025 
43 32 
g RTN 

007 
45 1 
RCL 1 

026 
42,21,13 
f LBL C 

008 
32 13 
GSB C 

027 
32 12 
GSB B 

009 
45 2 
RCL 2 

028 
43,30, 2 
g TEST x<0 

010 
32 13 
GSB C 

029 
32 14 
GSB D 

011 
40 
+ 

030 
3 
3 

012 
45 3 
RCL 3 

031 
14 
yˣ 

013 
32 13 
GSB C 

032 
43 32 
g RTN 

014 
40 
+ 

033 
42,21,14 
f LBL D 

015 
32 12 
GSB B 

034 
26 
EEX 

016 
43 32 
g RTN 

035 
3 
3 

017 
42,21,12 
f LBL B 

036 
40 
+ 

018 
26 
EEX 

037 
43 32 
g RTN 
