Pseudo Diophantine equation - sum of 3 cubes
J E Patterson - jepspectro.com - 20231119
Description
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 hp-15c is exceeded so this was calculated using Thomas Okken's Free42 simulator.
Another option is to use Wolfram Alpha. For example 751458175153 + (100000000000 - 38812075974)3 + 232973356313 = 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
804357581458175153−805387388120759743+126021232973356313 = 42
7515 ENTER
5974 CHS ENTER
5631 GSB A
42 is displayed
23+23-13 = 15
2 ENTER
2 ENTER
1 CHS GSB A
15 is displayed
5699368212219623807203−5699368211135634935093−4727154934533270323 = 3
720 ENTER
493509 CHS ENTER
7032 CHS GSB A
3 is displayed
88661289752875283−87784054428622393−27361114688070403 = 33
7528 ENTER
62239 CHS ENTER
807040 CHS GSB A
33 is displayed
88661289752875283−87784054428622393−27361114688070403 = 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 |
|