## To make {(x+1),(y+1),(x+y+1),(x-y+1)} all square

Find two integers   $x, \; y$   such that

$x \; + \; 1$
$y \; + \; 1$
$x \; + \; y \; + \; 1$
$x \; - \; y \; + \; 1$

are all squares

Here’s one example,     $x = 168$,    $y = 120$

$168 \; + \; 1 \; = \; 13^2$
$120 \; + \; 1 \; = \; 11^2$
$168 \; + \; 120 \; + \; 1 \; = \; 17^2$
$168 \; - \; 120 \; + \; 1 \; = \; 7^2$

Can you find another pair   $(x, \; y)$ ?

math grad - Interest: Number theory
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### 4 Responses to To make {(x+1),(y+1),(x+y+1),(x-y+1)} all square

1. paul says:

Here are 3 more

x = 1155, y = 960
1155 + 1 = 34^2
960 + 1 = 31^2
1155 + 960 + 1 = 46^2
1155 – 960 + 1 = 14^2

x = 1368, y = 840
1368 + 1 = 37^2
840 + 1 = 29^2
1368 + 840 + 1 = 47^2
1368 – 840 + 1 = 23^2

x = 2499, y = 2400
2499 + 1 = 50^2
2400 + 1 = 49^2
2499 + 2400 + 1 = 70^2
2499 – 2400 + 1 = 10^2

Paul.

• paul says:

and those where Y < x < 1000000

x = 5624, y = 3024
5624 + 1 = 75^2
3024 + 1 = 55^2
5624 + 3024 + 1 = 93^2
5624 – 3024 + 1 = 51^2

x = 7920, y = 6240
7920 + 1 = 89^2
6240 + 1 = 79^2
7920 + 6240 + 1 = 119^2
7920 – 6240 + 1 = 41^2

x = 29928, y = 17160
29928 + 1 = 173^2
17160 + 1 = 131^2
29928 + 17160 + 1 = 217^2
29928 – 17160 + 1 = 113^2

x = 33488, y = 11880
33488 + 1 = 183^2
11880 + 1 = 109^2
33488 + 11880 + 1 = 213^2
33488 – 11880 + 1 = 147^2

x = 47523, y = 43680
47523 + 1 = 218^2
43680 + 1 = 209^2
47523 + 43680 + 1 = 302^2
47523 – 43680 + 1 = 62^2

x = 54288, y = 43680
54288 + 1 = 233^2
43680 + 1 = 209^2
54288 + 43680 + 1 = 313^2
54288 – 43680 + 1 = 103^2

x = 60515, y = 25920
60515 + 1 = 246^2
25920 + 1 = 161^2
60515 + 25920 + 1 = 294^2
60515 – 25920 + 1 = 186^2

x = 70224, y = 63000
70224 + 1 = 265^2
63000 + 1 = 251^2
70224 + 63000 + 1 = 365^2
70224 – 63000 + 1 = 85^2

x = 81795, y = 58080
81795 + 1 = 286^2
58080 + 1 = 241^2
81795 + 58080 + 1 = 374^2
81795 – 58080 + 1 = 154^2

x = 88803, y = 57120
88803 + 1 = 298^2
57120 + 1 = 239^2
88803 + 57120 + 1 = 382^2
88803 – 57120 + 1 = 178^2

x = 155235, y = 43680
155235 + 1 = 394^2
43680 + 1 = 209^2
155235 + 43680 + 1 = 446^2
155235 – 43680 + 1 = 334^2

x = 245024, y = 235224
245024 + 1 = 495^2
235224 + 1 = 485^2
245024 + 235224 + 1 = 693^2
245024 – 235224 + 1 = 99^2

x = 310248, y = 175560
310248 + 1 = 557^2
175560 + 1 = 419^2
310248 + 175560 + 1 = 697^2
310248 – 175560 + 1 = 367^2

x = 372099, y = 297024
372099 + 1 = 610^2
297024 + 1 = 545^2
372099 + 297024 + 1 = 818^2
372099 – 297024 + 1 = 274^2

x = 422499, y = 93024
422499 + 1 = 650^2
93024 + 1 = 305^2
422499 + 93024 + 1 = 718^2
422499 – 93024 + 1 = 574^2

x = 422499, y = 201600
422499 + 1 = 650^2
201600 + 1 = 449^2
422499 + 201600 + 1 = 790^2
422499 – 201600 + 1 = 470^2

x = 616224, y = 561000
616224 + 1 = 785^2
561000 + 1 = 749^2
616224 + 561000 + 1 = 1085^2
616224 – 561000 + 1 = 235^2

Paul.

• paul says:

and why not to 10000000

x = 1056783, y = 261120
1056783 + 1 = 1028^2
261120 + 1 = 511^2
1056783 + 261120 + 1 = 1148^2
1056783 – 261120 + 1 = 892^2

x = 1085763, y = 982080
1085763 + 1 = 1042^2
982080 + 1 = 991^2
1085763 + 982080 + 1 = 1438^2
1085763 – 982080 + 1 = 322^2

x = 1505528, y = 844560
1505528 + 1 = 1227^2
844560 + 1 = 919^2
1505528 + 844560 + 1 = 1533^2
1505528 – 844560 + 1 = 813^2

x = 1625624, y = 491400
1625624 + 1 = 1275^2
491400 + 1 = 701^2
1625624 + 491400 + 1 = 1455^2
1625624 – 491400 + 1 = 1065^2

x = 1755624, y = 303600
1755624 + 1 = 1325^2
303600 + 1 = 551^2
1755624 + 303600 + 1 = 1435^2
1755624 – 303600 + 1 = 1205^2

x = 1999395, y = 776160
1999395 + 1 = 1414^2
776160 + 1 = 881^2
1999395 + 776160 + 1 = 1666^2
1999395 – 776160 + 1 = 1106^2

x = 2550408, y = 2042040
2550408 + 1 = 1597^2
2042040 + 1 = 1429^2
2550408 + 2042040 + 1 = 2143^2
2550408 – 2042040 + 1 = 713^2

x = 2829123, y = 2825760
2829123 + 1 = 1682^2
2825760 + 1 = 1681^2
2829123 + 2825760 + 1 = 2378^2
2829123 – 2825760 + 1 = 58^2

x = 2873024, y = 2449224
2873024 + 1 = 1695^2
2449224 + 1 = 1565^2
2873024 + 2449224 + 1 = 2307^2
2873024 – 2449224 + 1 = 651^2

x = 3125823, y = 591360
3125823 + 1 = 1768^2
591360 + 1 = 769^2
3125823 + 591360 + 1 = 1928^2
3125823 – 591360 + 1 = 1592^2

x = 3330624, y = 491400
3330624 + 1 = 1825^2
491400 + 1 = 701^2
3330624 + 491400 + 1 = 1955^2
3330624 – 491400 + 1 = 1685^2

x = 4443663, y = 3690240
4443663 + 1 = 2108^2
3690240 + 1 = 1921^2
4443663 + 3690240 + 1 = 2852^2
4443663 – 3690240 + 1 = 868^2

x = 4605315, y = 2825760
4605315 + 1 = 2146^2
2825760 + 1 = 1681^2
4605315 + 2825760 + 1 = 2726^2
4605315 – 2825760 + 1 = 1334^2

x = 5424240, y = 5340720
5424240 + 1 = 2329^2
5340720 + 1 = 2311^2
5424240 + 5340720 + 1 = 3281^2
5424240 – 5340720 + 1 = 289^2

x = 6105840, y = 5997600
6105840 + 1 = 2471^2
5997600 + 1 = 2449^2
6105840 + 5997600 + 1 = 3479^2
6105840 – 5997600 + 1 = 329^2

x = 7425624, y = 4182024
7425624 + 1 = 2725^2
4182024 + 1 = 2045^2
7425624 + 4182024 + 1 = 3407^2
7425624 – 4182024 + 1 = 1801^2

x = 8185320, y = 8128200
8185320 + 1 = 2861^2
8128200 + 1 = 2851^2
8185320 + 8128200 + 1 = 4039^2
8185320 – 8128200 + 1 = 239^2

x = 9796899, y = 1560000
9796899 + 1 = 3130^2
1560000 + 1 = 1249^2
9796899 + 1560000 + 1 = 3370^2
9796899 – 1560000 + 1 = 2870^2

Paul.

2. benvitalis says:

$x = a^2 - 1$
$y = b^2 - 1$
$x + y = c^2 - 1$
$x - y = d^2 - 1$
$2 \,x = c^2 + d^2 - 2$
$2 \,a^2 - 2 = c^2 + d^2 - 2$
$2 \,a^2 = c^2 + d^2$
c, a, d are in A.P.