## 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

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)

math grad - Interest: Number theory
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### 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.