## 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

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

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

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

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

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

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

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

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

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

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

math grad - Interest: Number theory
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### 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