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
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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.

  2. Paul says:

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

  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

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