Integers (a,b,c,d); each pairwise sum & sum of all four is a square

 
 
Find four integers   (a, \; b, \; c, \; d)   such that

a \; + \; b
a \; + \; c
a \; + \; d
b \; + \; c
b \; + \; d
c \; + \; d
a \; + \; b \; + \; c \; + \; d
 
are all squares

 

The first few solutions are:

\{ \, 386, \; 2114, \; 3970, \; 10430 \, \}
\{ \, 590, \; 4594, \; 5810, \; 17906 \, \}
\{ \, 617, \; 15008, \; 26608, \; 63392 \, \}
\{ \, 872, \; 2377, \; 9944, \; 21032 \, \}
\{ \, 2248, \; 4808, \; 12881, \; 22088 \, \}

Also found by pipo

 
\{ \, 386, \; 2114, \; 3970, \; 10430 \, \}

386 + 2114 = 50^2
386 + 3970 = 66^2
386 + 10430 = 104^2
2114 + 3970 = 78^2
2114 + 10430 = 112^2
3970 + 10430 = 120^2
386 + 2114 + 3970 + 10430 = 130^2

\{ \, 590, \; 4594, \; 5810, \; 17906 \, \}

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

\{ \, 617, \; 15008, \; 26608, \; 63392 \, \}

617 + 15008 = 125^2
617 + 26608 = 165^2
617 + 63392 = 253^2
15008 + 26608 = 204^2
15008 + 63392 = 280^2
26608 + 63392 = 300^2
617 + 15008 + 26608 + 63392 = 325^2

\{ \, 872, \; 2377, \; 9944, \; 21032 \, \}

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

\{ \, 2248, \; 4808, \; 12881, \; 22088 \, \}

2248 + 4808 = 84^2
2248 + 12881 = 123^2
2248 + 22088 = 156^2
4808 + 12881 = 133^2
4808 + 22088 = 164^2
12881 + 22088 = 187^2
2248 + 4808 + 12881 + 22088 = 205^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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 Integers (a,b,c,d); each pairwise sum & sum of all four is a square

  1. pipo says:

    Here five solutions:
    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]
    [22088,12881,4808,2248,34969,26896,24336,17689,15129,7056,42025],[41720,15880,8456,1544,57600,50176,43264,24336,17424,10000,67600]
    The last one is not ‘primitive’ because it is 4 times the first solutions.

    pipo

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