## Each of {(a+b+1),(a+c+1),(b+c+1),(a+b+c+1)} is a square

Find positive integers   $a, b, c$   such that

$a + b + 1$
$a + c + 1$
$b + c + 1$
$a + b + c + 1$

are all square numbers

Here are few examples

$a \; < \; b \; < \; c \; < \; 1000$

$(a, \; b, \; c) \; = \; (231, \; 444, \; 924)$
$231 + 444 + 1 = 26^2$
$231 + 924 + 1 = 34^2$
$444 + 924 + 1 = 37^2$
$231 + 444 + 924 + 1 = 40^2$

Paul found:

$116 + 324 + 1 = 21^2$
$116 + 459 + 1 = 24^2$
$324 + 459 + 1 = 28^2$
$116 + 324 + 459 + 1 = 30^2$

pipo found:

$(a, \; b, \; c) \; = \; (67, \; 256, \; 832)$
$67 + 256 + 1 = 18^2$
$67 + 832 + 1 = 30^2$
$256 + 832 + 1 = 33^2$
$67 + 256 + 832 + 1 = 34^2$

$(a, \; b, \; c) \; = \; (296, \; 432, \; 792)$
$296 + 432 + 1 = 27^2$
$296 + 792 + 1 = 33^2$
$432 + 792 + 1 = 35^2$
$296 + 432 + 792 + 1 = 39^2$

See below for   $a \; < \; b \; < \; c \; < \; 10000$

math grad - Interest: Number theory
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### 7 Responses to Each of {(a+b+1),(a+c+1),(b+c+1),(a+b+c+1)} is a square

1. Paul says:

Here’s one, a < b < c 116, b ->324, c ->459, w ->21, x ->24, y ->28, z ->30

Paul.

• Paul says:

hmm that didn’t come out as I intended

a<b<c 116, b->324, c->459, w->21, x->24, y->28, z->30.

2. Paul says:

well it should say less than 500 and a = 116, etc

• benvitalis says:

• benvitalis says:

I found a triple for   $a \; < \; b \; < \; c \; < \; 1000$

3. pipo says:

Found two other solutions where a, b, c <1000
Format (a, b, c, sq(a+b+1), a+b+1, sq(a+c+1), a+c+1, sq(b+c+1), b+c+1, sq(a+b+c+1), a+b+c+1)
67 256 832 18 324 30 900 33 1089 34 1156
296 432 792 27 729 33 1089 35 1225 39 1521

And a lot more (first three squares <10000)
116 324 459 21 441 24 576 28 784 30 900
99 384 2016 22 484 46 2116 49 2401 50 2500
231 444 924 26 676 34 1156 37 1369 40 1600
291 384 1824 26 676 46 2116 47 2209 50 2500
188 540 1575 27 729 42 1764 46 2116 48 2304
267 516 1332 28 784 40 1600 43 1849 46 2116
228 555 2580 28 784 53 2809 56 3136 58 3364
159 624 5616 28 784 76 5776 79 6241 80 6400
216 624 2184 29 841 49 2401 53 2809 55 3025
387 512 3456 30 900 62 3844 63 3969 66 4356
440 648 2160 33 1089 51 2601 53 2809 57 3249
99 1056 1344 34 1156 38 1444 49 2401 50 2500
192 1032 1176 35 1225 37 1369 47 2209 49 2401
312 912 5016 35 1225 73 5329 77 5929 79 6241
315 980 1620 36 1296 44 1936 51 2601 54 2916
408 960 1440 37 1369 43 1849 49 2401 53 2809
552 816 3672 37 1369 65 4225 67 4489 71 5041
348 1020 6375 37 1369 82 6724 86 7396 88 7744
588 855 1260 38 1444 43 1849 46 2116 52 2704
447 1152 4176 40 1600 68 4624 73 5329 76 5776
423 1340 3420 42 1764 62 3844 69 4761 72 5184
332 1431 5292 42 1764 75 5625 82 6724 84 7056
252 1596 2247 43 1849 50 2500 62 3844 64 4096
708 1140 1995 43 1849 52 2704 56 3136 62 3844
744 1104 7176 43 1849 89 7921 91 8281 95 9025
159 1776 4464 44 1936 68 4624 79 6241 80 6400
480 1635 1728 46 2116 47 2209 58 3364 62 3844
396 1719 7884 46 2116 91 8281 98 9604 100 10000
576 1632 3267 47 2209 62 3844 70 4900 74 5476
648 1560 4680 47 2209 73 5329 79 6241 83 6889
756 1547 2052 48 2304 53 2809 60 3600 66 4356
1088 1215 2880 48 2304 63 3969 64 4096 72 5184
895 1408 6160 48 2304 84 7056 87 7569 92 8464
624 1776 3999 49 2401 68 4624 76 5776 80 6400
360 2040 5880 49 2401 79 6241 89 7921 91 8281
760 1840 6808 51 2601 87 7569 93 8649 97 9409
1152 1551 3072 52 2704 65 4225 68 4624 76 5776
1092 1611 6132 52 2704 85 7225 88 7744 94 8836
600 2208 3120 53 2809 61 3721 73 5329 77 5929
876 1932 2967 53 2809 62 3844 70 4900 76 5776
960 1848 4080 53 2809 71 5041 77 5929 83 6889
372 2436 6027 53 2809 80 6400 92 8464 94 8836
1068 1740 5655 53 2809 82 6724 86 7396 92 8464
1376 1539 5184 54 2916 81 6561 82 6724 90 8100
1032 1992 4896 55 3025 77 5929 83 6889 89 7921
1380 1755 2340 56 3136 61 3721 64 4096 74 5476
1359 1776 3264 56 3136 68 4624 71 5041 80 6400
328 2920 3640 57 3249 63 3969 81 6561 83 6889
688 2560 4495 57 3249 72 5184 84 7056 88 7744
1008 2240 4320 57 3249 73 5329 81 6561 87 7569
483 2880 3360 58 3364 62 3844 79 6241 82 6724
519 2844 4380 58 3364 70 4900 85 7225 88 7744
924 2556 2919 59 3481 62 3844 74 5476 80 6400
1296 2184 3744 59 3481 71 5041 77 5929 85 7225
1727 1872 3456 60 3600 72 5184 73 5329 84 7056
984 2736 3504 61 3721 67 4489 79 6241 85 7225
1728 2115 2240 62 3844 63 3969 66 4356 78 6084
672 3171 3552 62 3844 65 4225 82 6724 86 7396
1632 2211 7392 62 3844 95 9025 98 9604 106 11236
1224 2744 7056 63 3969 91 8281 99 9801 105 11025
1176 3312 5712 67 4489 83 6889 95 9025 101 10201
1520 3240 5040 69 4761 81 6561 91 8281 99 9801
2299 2884 6916 72 5184 96 9216 99 9801 110 12100
1152 4176 4472 73 5329 75 5625 93 8649 99 9801
2472 2856 6552 73 5329 95 9025 97 9409 109 11881
2492 3132 3591 75 5625 78 6084 82 6724 96 9216
2060 3564 6039 75 5625 90 8100 98 9604 108 11664
2400 3840 4995 79 6241 86 7396 94 8836 106 11236
3104 3456 4464 81 6561 87 7569 89 7921 105 11025
2520 4040 5760 81 6561 91 8281 99 9801 111 12321
4320 4515 5088 94 8836 97 9409 98 9604 118 13924

pipo

• benvitalis says:

Nice! I posted your solutions   $a \; < \; b \; < \; c \; < \; 1000$