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

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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.

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