a^n + b^n + c^n = x^n + y^n + z^n, n = 1,3

x^3 + y^3 + z^3 A1

x^3 + y^3 + z^3 A2

 
Find a few more solutions.

 
the smallest integer that can be expressed …

in 2 ways :     26
in 3 ways :     41
in 4 ways :     43

 

Paul found:

4 + 16 + 20 = 8 + 10 + 22 \; ....................... \; ( \,= 40 \,)
4^3 + 16^3 + 20^3 = 8^3 + 10^3 + 22^3 = 12160
 

1 + 17 + 23 = 5 + 11 + 25 \; ....................... \; ( \,= 41 \,)
1^3 + 17^3 + 23^3 = 5^3 + 11^3 + 25^3 = 17081

2 + 17 + 22 = 3 + 15 + 23 \; ....................... \; ( \,= 41 \,)
2^3 + 17^3 + 22^3 = 3^3 + 15^3 + 23^3 = 15569

5 + 16 + 20 = 6 + 14 + 21 \; ....................... \; ( \,= 41 \,)
5^3 + 16^3 + 20^3 = 6^3 + 14^3 + 21^3 = 12221
 

4 + 17 + 21 = 7 + 12 + 23 \; ....................... \; ( \,= 42 \,)
4^3 + 17^3 + 21^3 = 7^3 + 12^3 + 23^3 = 14238

4 + 18 + 20 = 9 + 10 + 23 \; ....................... \; ( \,= 42 \,)
4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3 = 13896

7 + 15 + 20 = 9 + 12 + 21 \; ....................... \; ( \,= 42 \,)
7^3 + 15^3 + 20^3 = 9^3 + 12^3 + 21^3 = 11718
 

1 + 19 + 23 = 3 + 15 + 25 \; ....................... \; ( \,= 43 \,)
1^3 + 19^3 + 23^3 = 3^3 + 15^3 + 25^3 = 19027

3 + 16 + 24 = 5 + 13 + 25 \; ....................... \; ( \,= 43 \,)
3^3 + 16^3 + 24^3 = 5^3 + 13^3 + 25^3 = 17947

5 + 15 + 23 = 8 + 11 + 24 \; ....................... \; ( \,= 43 \,)
5^3 + 15^3 + 23^3 = 8^3 + 11^3 + 24^3 = 15667

7 + 15 + 21 = 10 + 11 + 22 \; ...................... \; ( \,= 43 \,)
7^3 + 15^3 + 21^3 = 10^3 + 11^3 + 22^3 = 12979
 

1 + 20 + 25 = 7 + 11 + 28 \; ....................... \; ( \,= 46 \,)
1^3 + 20^3 + 25^3 = 7^3 + 11^3 + 28^3 = 23626

4 + 20 + 22 = 7 + 14 + 25 \; ....................... \; ( \,= 46 \,)
4^3 + 20^3 + 22^3 = 7^3 + 14^3 + 25^3 = 18712
 

5 + 17 + 25 = 7 + 14 + 26 \; ....................... \; ( \,= 47 \,)
5^3 + 17^3 + 25^3 = 7^3 + 14^3 + 26^3 = 20663

11 + 17 + 19 = 12 + 15 + 20 \; ..................... \; ( \,= 47 \,)
11^3 + 17^3 + 19^3 = 12^3 + 15^3 + 20^3 = 13103
 

3 + 16 + 29 = 8 + 10 + 30 \; ....................... \; ( \,= 48 \,)
3^3 + 16^3 + 29^3 = 8^3 + 10^3 + 30^3 = 28512

4 + 20 + 24 = 6 + 16 + 26 \; ....................... \; ( \,= 48 \,)
4^3 + 20^3 + 24^3 = 6^3 + 16^3 + 26^3 = 21888

8 + 18 + 22 = 9 + 16 + 23 \; ....................... \; ( \,= 48 \,)
8^3 + 18^3 + 22^3 = 9^3 + 16^3 + 23^3 = 16992
 

3 + 22 + 24 = 4 + 19 + 26 \; ....................... \; ( \,= 49 \,)
3^3 + 22^3 + 24^3 = 4^3 + 19^3 + 26^3 = 24499

4 + 17 + 28 = 7 + 13 + 29 \; ....................... \; ( \,= 49 \,)
4^3 + 17^3 + 28^3 = 7^3 + 13^3 + 29^3 = 26929

5 + 19 + 25 = 9 + 13 + 27 \; ....................... \; ( \,= 49 \,)
5^3 + 19^3 + 25^3 = 9^3 + 13^3 + 27^3 = 22609

7 + 19 + 23 = 10 + 14 + 25 \; ...................... \; ( \,= 49 \,)
7^3 + 19^3 + 23^3 = 10^3 + 14^3 + 25^3 = 19369
 

2 + 17 + 31 = 6 + 12 + 32 \; ....................... \; ( \,= 50 \,)
2^3 + 17^3 + 31^3 = 6^3 + 12^3 + 32^3 = 34712

2 + 20 + 28 = 6 + 14 + 30 \; ....................... \; ( \,= 50 \,)
2^3 + 20^3 + 28^3 = 6^3 + 14^3 + 30^3 = 29960

4 + 20 + 26 = 5 + 18 + 27 \; ....................... \; ( \,= 50 \,)
4^3 + 20^3 + 26^3 = 5^3 + 18^3 + 27^3 = 25640

 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

2 Responses to a^n + b^n + c^n = x^n + y^n + z^n, n = 1,3

  1. Paul says:

    Here is a list with 40<= a + b + c<=50

    4 + 16 + 20 = 8 + 10 + 22 = 40
    4^3 + 16^3 + 20^3 = 8^3 + 10^3 + 22^3 = 64000

    1 + 17 + 23 = 5 + 11 + 25 = 41
    1^3 + 17^3 + 23^3 = 5^3 + 11^3 + 25^3 = 68921
    2 + 17 + 22 = 3 + 15 + 23 = 41
    2^3 + 17^3 + 22^3 = 3^3 + 15^3 + 23^3 = 68921
    5 + 16 + 20 = 6 + 14 + 21 = 41
    5^3 + 16^3 + 20^3 = 6^3 + 14^3 + 21^3 = 68921

    4 + 17 + 21 = 7 + 12 + 23 = 42
    4^3 + 17^3 + 21^3 = 7^3 + 12^3 + 23^3 = 74088
    4 + 18 + 20 = 9 + 10 + 23 = 42
    4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3 = 74088
    7 + 15 + 20 = 9 + 12 + 21 = 42
    7^3 + 15^3 + 20^3 = 9^3 + 12^3 + 21^3 = 74088

    1 + 19 + 23 = 3 + 15 + 25 = 43
    1^3 + 19^3 + 23^3 = 3^3 + 15^3 + 25^3 = 79507
    3 + 16 + 24 = 5 + 13 + 25 = 43
    3^3 + 16^3 + 24^3 = 5^3 + 13^3 + 25^3 = 79507
    5 + 15 + 23 = 8 + 11 + 24 = 43
    5^3 + 15^3 + 23^3 = 8^3 + 11^3 + 24^3 = 79507
    7 + 15 + 21 = 10 + 11 + 22 = 43
    7^3 + 15^3 + 21^3 = 10^3 + 11^3 + 22^3 = 79507

    1 + 20 + 25 = 7 + 11 + 28 = 46
    1^3 + 20^3 + 25^3 = 7^3 + 11^3 + 28^3 = 97336
    4 + 20 + 22 = 7 + 14 + 25 = 46
    4^3 + 20^3 + 22^3 = 7^3 + 14^3 + 25^3 = 97336

    5 + 17 + 25 = 7 + 14 + 26 = 47
    5^3 + 17^3 + 25^3 = 7^3 + 14^3 + 26^3 = 103823
    11 + 17 + 19 = 12 + 15 + 20 = 47
    11^3 + 17^3 + 19^3 = 12^3 + 15^3 + 20^3 = 103823

    3 + 16 + 29 = 8 + 10 + 30 = 48
    3^3 + 16^3 + 29^3 = 8^3 + 10^3 + 30^3 = 110592
    4 + 20 + 24 = 6 + 16 + 26 = 48
    4^3 + 20^3 + 24^3 = 6^3 + 16^3 + 26^3 = 110592
    8 + 18 + 22 = 9 + 16 + 23 = 48
    8^3 + 18^3 + 22^3 = 9^3 + 16^3 + 23^3 = 110592

    3 + 22 + 24 = 4 + 19 + 26 = 49
    3^3 + 22^3 + 24^3 = 4^3 + 19^3 + 26^3 = 117649
    4 + 17 + 28 = 7 + 13 + 29 = 49
    4^3 + 17^3 + 28^3 = 7^3 + 13^3 + 29^3 = 117649
    5 + 19 + 25 = 9 + 13 + 27 = 49
    5^3 + 19^3 + 25^3 = 9^3 + 13^3 + 27^3 = 117649
    7 + 19 + 23 = 10 + 14 + 25 = 49
    7^3 + 19^3 + 23^3 = 10^3 + 14^3 + 25^3 = 117649

    2 + 17 + 31 = 6 + 12 + 32 = 50
    2^3 + 17^3 + 31^3 = 6^3 + 12^3 + 32^3 = 125000
    2 + 20 + 28 = 6 + 14 + 30 = 50
    2^3 + 20^3 + 28^3 = 6^3 + 14^3 + 30^3 = 125000
    4 + 20 + 26 = 5 + 18 + 27 = 50
    4^3 + 20^3 + 26^3 = 5^3 + 18^3 + 27^3 = 125000

    Paul.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s