Equations: A^2 + B^3 = C^4 and D^3 + E^4 = F^5

 
Find positive integer solutions to

(1)   A_1^2 \; \; + \; \; B_1^3 \; = \; C_1^4
(2)   A_2^3 \; \; + \; \; B_2^4 \; = \; C_2^5

 
For example,
 

(1)

 
3^3 \; = \; 27                              (3^3)^2 \; = \; 3^6
2 \; 3^2 \; = \; 18                          (2 3^2)^3 \; = \; 2^3 \; 3^6
3^6 \; + \; 2^3 \; 3^6
= 3^6 \; (1 + 2^3)
= 3^6 \; 3^2
= 3^8
= 9^4

                                                             27^2 \; + \; 18^3 \; = \; 6561 \; = \; 9^4

7 \; 3^2 \; = \; 63                            (7 \; 3^2)^2 \; = \; 7^2 \; 3^4
2^2 \; 3^2 \; = \; 36                          (2^2 3^2)^3 \; = \; 2^6 \; 3^6 \; = \; 2^6 \; 3^2 \; 3^4
7^2 \; 3^4 \; + \; 2^6 \; 3^2 \; 3^4
= 3^4 \; (7^2 + 2^6 3^2)
= 3^4 \; 5^4
= 15^4

                                                             63^2 \; + \; 36^3 \; = \; 50625 \; = \; 15^4

433   is a prime number.
143 = 11\; \times \: 13

                                                             433^2 \; + \; 143^3 \; = \; 3111696 \; = \; 42^4

 

                                                             28^2 \; + \; 8^3\; = \; 1296\; = \; 6^4

                                                             648^2 \; + \; 108^3\; = \; 1679616\; = \; 36^4

                                                             110592^2 \; + \; 4608^3\; = \; 110075314176\; = \; 576^4

 

                                                             ——————————————
 
(2)

 
2^8 \; = \; 256              (2^8)^3 \; = \; 2^{24}
2^6 \; = \; 64                 (2^6)^4 \; = \; 2^{24}
2^5 \; = \; 32

2^{24}\; +\; 2^{24} \; = \; 2^{25}

                                                             256^3\; +\; 64^4 \; = \; 32^5

 

Note that     7^{15} \; + \; 7^{16} \; = \; 7^{15} \; (1+7) \; = \; 7^{15} \; (2^3)

2^4 \; 7^5 \; = \; 268912              (2^4 \; 7^5)^3 \; = \; 2^{12} \; 7^{15}
2^3 \; 7^4 \; = \; 19208                 (2^3 \; 7^4)^4 \; = \; 2^{12} \; 7^{16}

2^{12} \; 7^{15} \; + \; 2^{12} \; 7^{16}
= \; 2^{12} \; (7^{15} + 7^{16})
= \; 2^{12} \; 7^{15} \; (2^3)
= \; 2^{15} \; 7^{15}
= \; (14^3)^5
= \; 2744^5

                                                             268912^3 \; + \; 19208^4 \; = \; 2744^5
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

1 Response to Equations: A^2 + B^3 = C^4 and D^3 + E^4 = F^5

  1. Paul says:
    Here are some solutions to part (1). $latex 4

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