perplexus.info :: Just Math : Sum Cubes and Fourth Powers

Sum Cubes and Fourth Powers (Posted on 2022-02-02)

x and y are two distinct positive integers such that:

(i) Their sum of cubes is a perfect square, and:
(ii) Their sum of fourth powers is a perfect cube.

Find the minimum value of x+y

See The Solution Submitted by K Sengupta

Rating: 4.0000 (1 votes)

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computer solution

| Comment 2 of 4 |

for tot=3:10000

for x=1:floor(tot/2)

y=tot-x;

cubes=x^3+y^3;

sr=round(sqrt(cubes));

if sr*sr==cubes

fourths=x^4+y^4;

cr=round(fourths^(1/3));

if cr^3==fourths

fprintf('%5d %5d %12d %7d %20d %5d ',x,y,cubes,sr,fourths,cr)

end

finds

   x     y      sum of     square           sum of     cube
                 cubes      root        fourth powers  root
                            
   32    32        65536     256              2097152   128 
  289   578    217238121   14739         118587876497  4913 
 2048  2048  17179869184  131072       35184372088832 32768

In the first case x and y are not distinct, so that the minimum x+y is 289 + 578 = 867.

Edited on February 2, 2022, 11:26 am

Posted by Charlie on 2022-02-02 11:23:58

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