a^5 – b^5 = c^3 – d^3 = e^2 – f^2

 
 
Find solutions to
 
a^5 \; - \; b^5 \; = \; c^3 \; - \; d^3 \; = \; e^2 \; - \; f^2

where   a, b, c, d, e, f   are positive integers.

 

for example,

4^5 \; - \; 2^5 \; = \; 10^3 \; - \; 2^3 \; = \; 39^2 \; - \; 23^2 \; = \; 992

Note that there are other solutions for the difference of squares:

992 \; = \; 66^2 \; - \; 58^2
992 \; = \; 126^2 \; - \; 122^2
992 \; = \; 249^2 \; - \; 247^2

 
 
Find other solutions

 
 

Paul found:

6^5 \; - \; 3^5 \; = \; 21^3 \; - \; 12^3 \; = \; 87^2 \; - \; 6^2
8^5 \; - \; 1^5 \; = \; 32^3 \; - \; 1^3 \; = \; 184^2 \; - \; 33^2
9^5 \; - \; 6^5 \; = \; 48^3 \; - \; 39^3 \; = \; 227^2 \; - \; 16^2
14^5 \; - \; 7^5 \; = \; 161^3 \; - \; 154^3 \; = \; 931^2 \; - \; 588^2

 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

2 Responses to a^5 – b^5 = c^3 – d^3 = e^2 – f^2

  1. paul says:

    This formatting might turn out nasty, the format can be seen if you follow the first one against the example.

    {4,2} ,
    10 2 39 23
    10 2 66 58
    10 2 126 122
    10 2 249 247
    992

    {6,3} ,
    21 12 87 6
    21 12 137 106
    21 12 153 126
    21 12 423 414
    21 12 1257 1254
    21 12 3767 3766
    7533

    {8,1} ,
    32 1 184 33
    32 1 544 513
    32 1 2344 2337
    32 1 16384 16383
    32767

    {9,6} ,
    48 39 227 16
    48 39 357 276
    48 39 963 936
    48 39 2853 2844
    48 39 8547 8544
    48 39 25637 25636
    51273

    {14,7} ,
    161 154 931 588
    161 154 1309 1092
    161 154 5341 5292
    161 154 8419 8388
    161 154 37219 37212
    161 154 260509 260508
    521017

    Paul.

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