Equation : A^2 + B^2 + 1 = C^2 + D^2

Let’s find solutions to

$A^2 \; + \; B^2 \; + \; 1 \; = \; C^2 \; + \; D^2$

Note that

$(n + 1)^2 \; + \; (2 \, n)^2 \; + \; 1$
$= \; 5 \, n^2 \; + \; 2 \, n \; + \; 2$
$= \; (2 \, n + 1)^2 \; + \; (n - 1)^2$

$2^2 \; + \; 2^2 \; + \; 1 \; = \; 9 \; = \; 3^2 \; + \; 0^2$
$3^2 \; + \; 4^2 \; + \; 1 \; = \; 26 \; = \; 5^2 \; + \; 1^2$
$4^2 \; + \; 6^2 \; + \; 1 \; = \; 53 \; = \; 7^2 \; + \; 2^2$
$5^2 \; + \; 8^2 \; + \; 1 \; = \; 90 \; = \; 9^2 \; + \; 3^2$
$6^2 \; + \; 10^2 \; + \; 1 \; = \; 137 \; = \; 11^2 \; + \; 4^2$

Let’s find   $(x, y)$   so that

$(x + 1)^2 \; + \; (2 \, x)^2 \; + \; 1 \; = \; (2 \, x + 1)^2 \; + \; (x - 1)^2 \; = \; y^2$

Here are the first few solutions:

$2^2 + 2^2 + 1 = 3^2 = 3^2 + 0^2$

$(409+1)^2 + (2\times 409)^2 + 1$
$= (2\times 409 + 1)^2 + (409 - 1)^2$
$= 837225$
$= 915^2$

$(131761 + 1)^2 + (2\times 131761)^2 + 1$
$= (2\times 131761 + 1)^2 + (131761 - 1)^2$
$= 86805069129$
$= 294627^2$

$(42426697 + 1)^2 + (2\times 42426697)^2 + 1$
$= (2\times 42426697 + 1)^2 + (42426697 - 1)^2$
$= 9000123176502441$
$= 94868979^2$

$(13661264737 + 1)^2 + (2\times 13661264737)^2 + 1$
$= (2\times 13661264737 + 1)^2 + (13661264737 - 1)^2$
$= 933150771099320925321$
$= 30547516611^2$

$(4398884818681 + 1)^2 + (2\times 4398884818681)^2 + 1$
$= (2\times 4398884818681 + 1)^2 + (4398884818681 - 1)^2$
$= 96750938240119669002536169$
$= 9836205479763^2$

$(1416427250350609 + 1)^2 + (2\times 1416427250350609)^2 + 1$
$= (2\times 1416427250350609 + 1)^2 + (1416427250350609 - 1)^2$
$= 10031330777678936750421634055625$
$= 3167227616967075^2$

$(456085175728077481 + 1)^2 + (2\times 456085175728077481)^2 + 1$
$= (2\times 456085175728077481 + 1)^2 + (456085175728077481 - 1)^2$
$= 1040068437594556581917096193152681769$
$= 1019837456457918387^2$

$(146858010157190598337 + 1)^2 + (2\times 146858010157190598337)^2 + 1$
$= (2\times 146858010157190598337 + 1)^2 + (146858010157190598337 - 1)^2$
$= 107836375736647484748649430748034717024521$
$= 328384493751832753539^2$

$(47287823185439644587097 + 1)^2 + (2\times 47287823185439644587097)^2 + 1$
$= (2\times 47287823185439644587097 + 1)^2 + (47287823185439644587097 - 1)^2$
$= 11180691108087016076114913696900639337383611241$
$= 105738787150633688721171^2$

$(15226532207701408366446961 + 1)^2 + (2\times 15226532207701408366446961)^2 + 1$
$= (2\times 15226532207701408366446961 + 1)^2 + (15226532207701408366446961 - 1)^2$
$= 1159236415360841625067098997173402657030572863571529$
$= 34047561078010295935463523^2$

………………………

Find other solutions to    $A^2 \; + \; B^2 \; + \; 1 \; = \; C^2 \; + \; D^2$

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

One Response to Equation : A^2 + B^2 + 1 = C^2 + D^2

1. paul says:

Here are those solutions where a<=b<50 and x<=y<=50

1^2 + 4^2 + 1 = 3^2 + 3^2
1^2 + 12^2 + 1 = 5^2 + 11^2
1^2 + 24^2 + 1 = 7^2 + 23^2
1^2 + 24^2 + 1 = 17^2 + 17^2
1^2 + 40^2 + 1 = 9^2 + 39^2
1^2 + 48^2 + 1 = 25^2 + 41^2
2^2 + 6^2 + 1 = 4^2 + 5^2
2^2 + 12^2 + 1 = 7^2 + 10^2
2^2 + 16^2 + 1 = 6^2 + 15^2
2^2 + 20^2 + 1 = 9^2 + 18^2
2^2 + 30^2 + 1 = 8^2 + 29^2
2^2 + 30^2 + 1 = 11^2 + 28^2
2^2 + 36^2 + 1 = 25^2 + 26^2
2^2 + 42^2 + 1 = 13^2 + 40^2
2^2 + 42^2 + 1 = 20^2 + 37^2
2^2 + 48^2 + 1 = 10^2 + 47^2
3^2 + 4^2 + 1 = 1^2 + 5^2
3^2 + 8^2 + 1 = 5^2 + 7^2
3^2 + 20^2 + 1 = 7^2 + 19^2
3^2 + 20^2 + 1 = 11^2 + 17^2
3^2 + 24^2 + 1 = 15^2 + 19^2
3^2 + 28^2 + 1 = 13^2 + 25^2
3^2 + 36^2 + 1 = 9^2 + 35^2
3^2 + 48^2 + 1 = 17^2 + 45^2
3^2 + 48^2 + 1 = 33^2 + 35^2
4^2 + 6^2 + 1 = 2^2 + 7^2
4^2 + 9^2 + 1 = 7^2 + 7^2
4^2 + 10^2 + 1 = 6^2 + 9^2
4^2 + 15^2 + 1 = 11^2 + 11^2
4^2 + 17^2 + 1 = 9^2 + 15^2
4^2 + 21^2 + 1 = 13^2 + 17^2
4^2 + 24^2 + 1 = 8^2 + 23^2
4^2 + 27^2 + 1 = 11^2 + 25^2
4^2 + 28^2 + 1 = 15^2 + 24^2
4^2 + 36^2 + 1 = 17^2 + 32^2
4^2 + 36^2 + 1 = 23^2 + 28^2
4^2 + 39^2 + 1 = 13^2 + 37^2
4^2 + 42^2 + 1 = 10^2 + 41^2
4^2 + 42^2 + 1 = 25^2 + 34^2
4^2 + 45^2 + 1 = 19^2 + 41^2
5^2 + 8^2 + 1 = 3^2 + 9^2
5^2 + 12^2 + 1 = 1^2 + 13^2
5^2 + 12^2 + 1 = 7^2 + 11^2
5^2 + 28^2 + 1 = 9^2 + 27^2
5^2 + 36^2 + 1 = 19^2 + 31^2
5^2 + 44^2 + 1 = 21^2 + 39^2
5^2 + 48^2 + 1 = 11^2 + 47^2
5^2 + 48^2 + 1 = 31^2 + 37^2
6^2 + 6^2 + 1 = 3^2 + 8^2
6^2 + 8^2 + 1 = 1^2 + 10^2
6^2 + 10^2 + 1 = 4^2 + 11^2
6^2 + 12^2 + 1 = 9^2 + 10^2
6^2 + 14^2 + 1 = 8^2 + 13^2
6^2 + 16^2 + 1 = 2^2 + 17^2
6^2 + 22^2 + 1 = 11^2 + 20^2
6^2 + 24^2 + 1 = 17^2 + 18^2
6^2 + 28^2 + 1 = 14^2 + 25^2
6^2 + 30^2 + 1 = 19^2 + 24^2
6^2 + 32^2 + 1 = 10^2 + 31^2
6^2 + 34^2 + 1 = 13^2 + 32^2
6^2 + 38^2 + 1 = 16^2 + 35^2
6^2 + 40^2 + 1 = 26^2 + 31^2
6^2 + 42^2 + 1 = 24^2 + 35^2
6^2 + 44^2 + 1 = 23^2 + 38^2
6^2 + 46^2 + 1 = 28^2 + 37^2
6^2 + 48^2 + 1 = 15^2 + 46^2
7^2 + 12^2 + 1 = 5^2 + 13^2
7^2 + 16^2 + 1 = 9^2 + 15^2
7^2 + 20^2 + 1 = 3^2 + 21^2
7^2 + 20^2 + 1 = 15^2 + 15^2
7^2 + 24^2 + 1 = 1^2 + 25^2
7^2 + 36^2 + 1 = 11^2 + 35^2
8^2 + 9^2 + 1 = 5^2 + 11^2
8^2 + 13^2 + 1 = 3^2 + 15^2
8^2 + 14^2 + 1 = 6^2 + 15^2
8^2 + 15^2 + 1 = 1^2 + 17^2
8^2 + 15^2 + 1 = 11^2 + 13^2
8^2 + 18^2 + 1 = 10^2 + 17^2
8^2 + 22^2 + 1 = 15^2 + 18^2
8^2 + 24^2 + 1 = 4^2 + 25^2
8^2 + 27^2 + 1 = 13^2 + 25^2
8^2 + 30^2 + 1 = 2^2 + 31^2
8^2 + 30^2 + 1 = 17^2 + 26^2
8^2 + 33^2 + 1 = 23^2 + 25^2
8^2 + 36^2 + 1 = 20^2 + 31^2
8^2 + 39^2 + 1 = 19^2 + 35^2
8^2 + 39^2 + 1 = 25^2 + 31^2
8^2 + 40^2 + 1 = 12^2 + 39^2
8^2 + 40^2 + 1 = 24^2 + 33^2
8^2 + 41^2 + 1 = 15^2 + 39^2
8^2 + 49^2 + 1 = 21^2 + 45^2
9^2 + 12^2 + 1 = 1^2 + 15^2
9^2 + 16^2 + 1 = 7^2 + 17^2
9^2 + 16^2 + 1 = 13^2 + 13^2
9^2 + 20^2 + 1 = 11^2 + 19^2
9^2 + 28^2 + 1 = 5^2 + 29^2
9^2 + 36^2 + 1 = 3^2 + 37^2
9^2 + 36^2 + 1 = 17^2 + 33^2
9^2 + 40^2 + 1 = 1^2 + 41^2
9^2 + 40^2 + 1 = 29^2 + 29^2
9^2 + 44^2 + 1 = 13^2 + 43^2
9^2 + 48^2 + 1 = 19^2 + 45^2
10^2 + 12^2 + 1 = 7^2 + 14^2
10^2 + 18^2 + 1 = 5^2 + 20^2
10^2 + 18^2 + 1 = 8^2 + 19^2
10^2 + 18^2 + 1 = 13^2 + 16^2
10^2 + 22^2 + 1 = 3^2 + 24^2
10^2 + 22^2 + 1 = 12^2 + 21^2
10^2 + 24^2 + 1 = 1^2 + 26^2
10^2 + 32^2 + 1 = 6^2 + 33^2
10^2 + 32^2 + 1 = 15^2 + 30^2
10^2 + 42^2 + 1 = 4^2 + 43^2
10^2 + 42^2 + 1 = 29^2 + 32^2
10^2 + 48^2 + 1 = 2^2 + 49^2
10^2 + 48^2 + 1 = 14^2 + 47^2
10^2 + 48^2 + 1 = 17^2 + 46^2
10^2 + 48^2 + 1 = 31^2 + 38^2
11^2 + 20^2 + 1 = 9^2 + 21^2
11^2 + 24^2 + 1 = 13^2 + 23^2
11^2 + 36^2 + 1 = 7^2 + 37^2
11^2 + 48^2 + 1 = 5^2 + 49^2
12^2 + 12^2 + 1 = 8^2 + 15^2
12^2 + 13^2 + 1 = 5^2 + 17^2
12^2 + 15^2 + 1 = 3^2 + 19^2
12^2 + 15^2 + 1 = 9^2 + 17^2
12^2 + 16^2 + 1 = 1^2 + 20^2
12^2 + 20^2 + 1 = 4^2 + 23^2
12^2 + 20^2 + 1 = 16^2 + 17^2
12^2 + 21^2 + 1 = 15^2 + 19^2
12^2 + 22^2 + 1 = 2^2 + 25^2
12^2 + 22^2 + 1 = 10^2 + 23^2
12^2 + 23^2 + 1 = 7^2 + 25^2
12^2 + 26^2 + 1 = 14^2 + 25^2
12^2 + 28^2 + 1 = 20^2 + 23^2
12^2 + 29^2 + 1 = 5^2 + 31^2
12^2 + 29^2 + 1 = 19^2 + 25^2
12^2 + 33^2 + 1 = 3^2 + 35^2
12^2 + 34^2 + 1 = 25^2 + 26^2
12^2 + 35^2 + 1 = 1^2 + 37^2
12^2 + 35^2 + 1 = 23^2 + 29^2
12^2 + 37^2 + 1 = 17^2 + 35^2
12^2 + 39^2 + 1 = 21^2 + 35^2
12^2 + 40^2 + 1 = 8^2 + 41^2
12^2 + 40^2 + 1 = 28^2 + 31^2
12^2 + 43^2 + 1 = 25^2 + 37^2
12^2 + 44^2 + 1 = 20^2 + 41^2
13^2 + 24^2 + 1 = 11^2 + 25^2
13^2 + 28^2 + 1 = 15^2 + 27^2
13^2 + 36^2 + 1 = 25^2 + 29^2
13^2 + 44^2 + 1 = 9^2 + 45^2
13^2 + 48^2 + 1 = 25^2 + 43^2
14^2 + 18^2 + 1 = 11^2 + 20^2
14^2 + 24^2 + 1 = 17^2 + 22^2
14^2 + 26^2 + 1 = 12^2 + 27^2
14^2 + 28^2 + 1 = 9^2 + 30^2
14^2 + 30^2 + 1 = 16^2 + 29^2
14^2 + 36^2 + 1 = 7^2 + 38^2
14^2 + 42^2 + 1 = 5^2 + 44^2
14^2 + 42^2 + 1 = 19^2 + 40^2
14^2 + 46^2 + 1 = 3^2 + 48^2
14^2 + 48^2 + 1 = 10^2 + 49^2
15^2 + 16^2 + 1 = 11^2 + 19^2
15^2 + 20^2 + 1 = 1^2 + 25^2
15^2 + 24^2 + 1 = 19^2 + 21^2
15^2 + 28^2 + 1 = 7^2 + 31^2
15^2 + 28^2 + 1 = 13^2 + 29^2
15^2 + 32^2 + 1 = 5^2 + 35^2
15^2 + 32^2 + 1 = 17^2 + 31^2
15^2 + 32^2 + 1 = 25^2 + 25^2
15^2 + 36^2 + 1 = 1^2 + 39^2
16^2 + 20^2 + 1 = 9^2 + 24^2
16^2 + 21^2 + 1 = 13^2 + 23^2
16^2 + 24^2 + 1 = 7^2 + 28^2
16^2 + 25^2 + 1 = 21^2 + 21^2
16^2 + 27^2 + 1 = 5^2 + 31^2
16^2 + 27^2 + 1 = 19^2 + 25^2
16^2 + 29^2 + 1 = 3^2 + 33^2
16^2 + 30^2 + 1 = 1^2 + 34^2
16^2 + 30^2 + 1 = 14^2 + 31^2
16^2 + 33^2 + 1 = 11^2 + 35^2
16^2 + 34^2 + 1 = 18^2 + 33^2
16^2 + 36^2 + 1 = 23^2 + 32^2
16^2 + 43^2 + 1 = 9^2 + 45^2
16^2 + 47^2 + 1 = 21^2 + 45^2
16^2 + 48^2 + 1 = 25^2 + 44^2
16^2 + 48^2 + 1 = 31^2 + 40^2
17^2 + 24^2 + 1 = 5^2 + 29^2
17^2 + 32^2 + 1 = 15^2 + 33^2
17^2 + 36^2 + 1 = 19^2 + 35^2
17^2 + 36^2 + 1 = 25^2 + 31^2
17^2 + 48^2 + 1 = 35^2 + 37^2
18^2 + 20^2 + 1 = 7^2 + 26^2
18^2 + 20^2 + 1 = 10^2 + 25^2
18^2 + 20^2 + 1 = 14^2 + 23^2
18^2 + 22^2 + 1 = 5^2 + 28^2
18^2 + 24^2 + 1 = 1^2 + 30^2
18^2 + 24^2 + 1 = 15^2 + 26^2
18^2 + 28^2 + 1 = 22^2 + 25^2
18^2 + 30^2 + 1 = 21^2 + 28^2
18^2 + 34^2 + 1 = 16^2 + 35^2
18^2 + 36^2 + 1 = 10^2 + 39^2
18^2 + 38^2 + 1 = 13^2 + 40^2
18^2 + 38^2 + 1 = 20^2 + 37^2
18^2 + 42^2 + 1 = 8^2 + 45^2
18^2 + 46^2 + 1 = 29^2 + 40^2
19^2 + 36^2 + 1 = 17^2 + 37^2
19^2 + 40^2 + 1 = 21^2 + 39^2
20^2 + 20^2 + 1 = 15^2 + 24^2
20^2 + 21^2 + 1 = 1^2 + 29^2
20^2 + 24^2 + 1 = 4^2 + 31^2
20^2 + 27^2 + 1 = 13^2 + 31^2
20^2 + 27^2 + 1 = 17^2 + 29^2
20^2 + 30^2 + 1 = 25^2 + 26^2
20^2 + 33^2 + 1 = 11^2 + 37^2
20^2 + 33^2 + 1 = 23^2 + 31^2
20^2 + 34^2 + 1 = 6^2 + 39^2
20^2 + 36^2 + 1 = 4^2 + 41^2
20^2 + 38^2 + 1 = 9^2 + 42^2
20^2 + 38^2 + 1 = 18^2 + 39^2
20^2 + 39^2 + 1 = 31^2 + 31^2
20^2 + 42^2 + 1 = 7^2 + 46^2
20^2 + 42^2 + 1 = 22^2 + 41^2
20^2 + 43^2 + 1 = 15^2 + 45^2
20^2 + 43^2 + 1 = 27^2 + 39^2
20^2 + 45^2 + 1 = 5^2 + 49^2
20^2 + 47^2 + 1 = 33^2 + 39^2
20^2 + 48^2 + 1 = 32^2 + 41^2
21^2 + 24^2 + 1 = 17^2 + 27^2
21^2 + 28^2 + 1 = 1^2 + 35^2
21^2 + 32^2 + 1 = 25^2 + 29^2
21^2 + 40^2 + 1 = 19^2 + 41^2
21^2 + 44^2 + 1 = 13^2 + 47^2
21^2 + 44^2 + 1 = 23^2 + 43^2
21^2 + 48^2 + 1 = 35^2 + 39^2
22^2 + 24^2 + 1 = 10^2 + 31^2
22^2 + 30^2 + 1 = 4^2 + 37^2
22^2 + 30^2 + 1 = 19^2 + 32^2
22^2 + 36^2 + 1 = 10^2 + 41^2
22^2 + 36^2 + 1 = 25^2 + 34^2
22^2 + 42^2 + 1 = 20^2 + 43^2
22^2 + 42^2 + 1 = 32^2 + 35^2
22^2 + 44^2 + 1 = 30^2 + 39^2
22^2 + 46^2 + 1 = 24^2 + 45^2
23^2 + 28^2 + 1 = 15^2 + 33^2
23^2 + 44^2 + 1 = 21^2 + 45^2
23^2 + 48^2 + 1 = 25^2 + 47^2
24^2 + 24^2 + 1 = 8^2 + 33^2
24^2 + 25^2 + 1 = 19^2 + 29^2
24^2 + 27^2 + 1 = 9^2 + 35^2
24^2 + 28^2 + 1 = 20^2 + 31^2
24^2 + 29^2 + 1 = 7^2 + 37^2
24^2 + 31^2 + 1 = 13^2 + 37^2
24^2 + 32^2 + 1 = 1^2 + 40^2
24^2 + 33^2 + 1 = 21^2 + 35^2
24^2 + 34^2 + 1 = 17^2 + 38^2
24^2 + 35^2 + 1 = 11^2 + 41^2
24^2 + 35^2 + 1 = 29^2 + 31^2
24^2 + 36^2 + 1 = 28^2 + 33^2
24^2 + 39^2 + 1 = 27^2 + 37^2
24^2 + 41^2 + 1 = 7^2 + 47^2
24^2 + 42^2 + 1 = 15^2 + 46^2
24^2 + 43^2 + 1 = 5^2 + 49^2
24^2 + 46^2 + 1 = 22^2 + 47^2
25^2 + 36^2 + 1 = 31^2 + 31^2
25^2 + 48^2 + 1 = 23^2 + 49^2
26^2 + 36^2 + 1 = 23^2 + 38^2
26^2 + 42^2 + 1 = 29^2 + 40^2
27^2 + 28^2 + 1 = 17^2 + 35^2
27^2 + 32^2 + 1 = 23^2 + 35^2
27^2 + 36^2 + 1 = 1^2 + 45^2
27^2 + 40^2 + 1 = 11^2 + 47^2
27^2 + 40^2 + 1 = 31^2 + 37^2
28^2 + 30^2 + 1 = 2^2 + 41^2
28^2 + 30^2 + 1 = 23^2 + 34^2
28^2 + 31^2 + 1 = 15^2 + 39^2
28^2 + 33^2 + 1 = 5^2 + 43^2
28^2 + 36^2 + 1 = 20^2 + 41^2
28^2 + 39^2 + 1 = 25^2 + 41^2
28^2 + 40^2 + 1 = 9^2 + 48^2
28^2 + 40^2 + 1 = 33^2 + 36^2
28^2 + 41^2 + 1 = 21^2 + 45^2
28^2 + 45^2 + 1 = 31^2 + 43^2
29^2 + 36^2 + 1 = 17^2 + 43^2
30^2 + 30^2 + 1 = 24^2 + 35^2
30^2 + 34^2 + 1 = 11^2 + 44^2
30^2 + 36^2 + 1 = 9^2 + 46^2
30^2 + 36^2 + 1 = 26^2 + 39^2
30^2 + 40^2 + 1 = 10^2 + 49^2
30^2 + 42^2 + 1 = 19^2 + 48^2
30^2 + 42^2 + 1 = 27^2 + 44^2
30^2 + 42^2 + 1 = 36^2 + 37^2
30^2 + 44^2 + 1 = 34^2 + 41^2
30^2 + 48^2 + 1 = 33^2 + 46^2
31^2 + 36^2 + 1 = 7^2 + 47^2
32^2 + 35^2 + 1 = 15^2 + 45^2
32^2 + 35^2 + 1 = 27^2 + 39^2
32^2 + 45^2 + 1 = 29^2 + 47^2
32^2 + 45^2 + 1 = 37^2 + 41^2
33^2 + 36^2 + 1 = 19^2 + 45^2
33^2 + 40^2 + 1 = 17^2 + 49^2
33^2 + 40^2 + 1 = 29^2 + 43^2
33^2 + 44^2 + 1 = 25^2 + 49^2
33^2 + 48^2 + 1 = 37^2 + 45^2
34^2 + 34^2 + 1 = 3^2 + 48^2
34^2 + 38^2 + 1 = 24^2 + 45^2
35^2 + 36^2 + 1 = 11^2 + 49^2
35^2 + 36^2 + 1 = 29^2 + 41^2
35^2 + 48^2 + 1 = 41^2 + 43^2
36^2 + 36^2 + 1 = 17^2 + 48^2
36^2 + 38^2 + 1 = 25^2 + 46^2
36^2 + 40^2 + 1 = 31^2 + 44^2
36^2 + 44^2 + 1 = 32^2 + 47^2
36^2 + 49^2 + 1 = 43^2 + 43^2
40^2 + 42^2 + 1 = 34^2 + 47^2
40^2 + 45^2 + 1 = 35^2 + 49^2
42^2 + 42^2 + 1 = 35^2 + 48^2

Paul.