Each of a+b,a+c,a+d,b+c,b+d,c+d,a+b+c+d is a square

 

 
Find four distinct integers   a, b, c, d   such that
a + b
a + c
a + d
b + c
b + d
c + d
a + b + c + d

are to be made squares

 

For example,     386,   2114,   3970,   10430

386 + 2114 = 2500 = 50^2
386 + 3970 = 4356 = 66^2
386 + 10430 = 10816 = 104^2
2114 + 3970 = 6084 = 78^2
2114 + 10430 = 12544 = 112^2
3970 + 10430 = 14400 = 120^2
386 + 2114 + 3970 + 10430 = 16900 = 130^2

 
 

(617, 15008, 26608, 63392)
617 + 15008 = 15625 = 125^2
617 + 26608 = 27225 = 165^2
617 + 63392 = 64009 = 253^2
15008 + 26608 = 41616 = 204^2
15008 + 63392 = 78400 = 280^2
26608 + 63392 = 90000 = 300^2
617 + 15008 + 26608 + 63392 = 105625 = 325^2

 
Allowing one integer to be negative, we have:

 
(-104, 360, 729, 3240)
-104 + 360 = 256 = 16^2
-104 + 729 = 625 = 25^2
-104 + 3240 = 3136 = 56^2
360 + 729 = 1089 = 33^2
360 + 3240 = 3600 = 60^2
729 + 3240 = 3969 = 63^2
-104 + 360 + 729 + 3240 = 4225 = 65^2

(-88, 344, 1177, 2792)
-88 + 344 = 256 = 16^2
-88 + 1177 = 1089 = 33^2
-88 + 2792 = 2704 = 52^2
344 + 1177 = 1521 = 39^2
344 + 2792 = 3136 = 56^2
1177 + 2792 = 3969 = 63^2
-88 + 344 + 1177 + 2792 = 4225 = 65^2

 

pipo found:

(17906,5810,4594,590)
(21032,9944,2377,872)
(22088,1881,4808,2248)

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

3 Responses to Each of a+b,a+c,a+d,b+c,b+d,c+d,a+b+c+d is a square

  1. pipo says:

    Found some more:
    Format (a,b,c,d,a+b,a+c,a+d,b+c,b+d,c+d,a+b+c+d)
    10430,3970,2114,386,14400,12544,10816,6084,4356,2500,16900
    17906,5810,4594,590,23716,22500,18496,10404,6400,5184,28900
    21032,9944,2377,872,30976,23409,21904,12321,10816,3249,34225
    22088,1881,4808,2248,34969,26896,24336,17689,15129,7056,42025
    41720,15880,8456,1544,57600,50176,43264,24336,17424,10000,67600
    The last one is 4 times the first one so not a primitive solution.

    pipo

  2. paul says:

    Here are 2 more

    {590,4594,5810,17906}
    590 + 4594 = 72^2
    590 + 5810 = 80^2
    590 + 17906 = 136^2
    4594 + 5810 = 102^2
    5810 + 17906 = 154^2
    4594 + 17906 = 150^2
    590 + 4594 + 5810 + 17906 = 170^2

    {872,2377,9944,21032}
    872 + 2377 = 57^2
    872 + 9944 = 104^2
    872 + 21032 = 148^2
    2377 + 9944 = 111^2
    9944 + 21032 = 176^2
    2377 + 21032 = 153^2
    872 + 2377 + 9944 + 21032 = 185^2

    Paul.

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