To make {(A + B^2), (A/n + B^2)} squares

 

A \; + \; B^2 \; = \; C^2
A/n \; + \; B^2 \; = \; D^2

 

n = 2

48 + 1 = 7^2 ......................... (48/2) + 1 = 5^2
1680 + 1 = 41^2 ...................... (1680/2) + 1 = 29^2
57120 + 1 = 239^2 .................... (57120/2) + 1 = 169^2
1940448 + 1 = 1393^2 ................. (1940448/2) + 1 = 985^2
65918160 + 1 = 8119^2 ................ (65918160/2) + 1 = 5741^2
2239277040 + 1 = 47321^2 ............. (2239277040/2) + 1 = 33461^2
76069501248 + 1 = 275807^2 ........... (76069501248/2) + 1 = 195025^2
2584123765440 + 1 = 1607521^2 ........ (2584123765440/2) + 1 = 1136689^2
……………………………………………………
……………………………………………………

Consider all Pythagorean triples   (x, x+1, z)   ordered by increasing   z
x^2 + (x+1)^2 = z^2
when   x^2 + (x+1)^2   doubled, we obtain,    (2 x+1)^2 + 1
then,    2 z^2 - 1 = (2 x+1)^2
the sequence   z,   5,   29,   169,   985,   5741,   33461,   195025,   1136689
  is such that
2 z^2   =   an odd square   +   1
 

n = 3

24 + 1 = 5^2 .................... (24/3) + 1 = 3^2
360 + 1 = 19^2 .................. (360/3) + 1 = 11^2
5040 + 1 = 71^2 ................. (5040/3) + 1 = 41^2
70224 + 1 = 265^2 ............... (70224/3) + 1 = 153^2
978120 + 1 = 989^2 .............. (978120/3) + 1 = 571^2
13623480 + 1 = 3691^2 ........... (13623480/3) + 1 = 2131^2
189750624 + 1 = 13775^2 ......... (189750624/3) + 1 = 7953^2
2642885280 + 1 = 51409^2 ........ (2642885280/3) + 1 = 29681^2
…………………………………………………..
…………………………………………………..

the sequence   3,   11,   41,   153,   571,   2131,   7953,   29681

is    a(n) = 4   a(n-1) – a(n-2)

 

n = 5

15 + 1 = 4^2 ................. (15/5) + 1 = 2^2 ............... = (F_{3})^2
120 + 1 = 11^2 ............... (120/5) + 1 = 5^2 .............. = (F_{5})^2
840 + 1 = 29^2 ............... (840/5) + 1 = 13^2 ............. = (F_{7})^2
5775 + 1 = 76^2 .............. (5775/5) + 1 = 34^2 ............ = (F_{9})^2
39600 + 1 = 199^2 ............ (39600/5) + 1 = 89^2 ........... = (F_{11})^2
271440 + 1 = 521^2 ........... (271440/5) + 1 = 233^2 ......... = (F_{13})^2
1860495 + 1 = 1364^2 ......... (1860495/5) + 1 = 610^2 ........ = (F_{15})^2
12752040 + 1 = 3571^2 ........ (12752040/5) + 1 = 1597^2 ...... = (F_{17})^2
87403800 + 1 = 9349^2 ........ (87403800/5) + 1 = 4181^2 ...... = (F_{19})^2
599074575 + 1 = 24476^2 ...... (599074575/5) + 1 = 10946^2 .... = (F_{21})^2
4106118240 + 1 = 64079^2 ..... (4106118240/5) + 1 = 28657^2 ... = (F_{23})^2
28143753120 + 1 = 167761^2 ... (28143753120/5) + 1 = 75025^2 .. = (F_{25})^2

 
n = 6

48 + 1 = 7^2 .................. (48/6) + 1 = 3^2
288 + 1 = 17^2 ................ (288/6) + 1 = 7^2
5040 + 1 = 71^2 ............... (5040/6) + 1 = 29^2
28560 + 1 = 169^2 ............. (28560/6) + 1 = 69^2
494208 + 1 = 703^2 ............ (494208/6) + 1 = 287^2
2798928 + 1 = 1673^2 .......... (2798928/6) + 1 = 683^2
48427680 + 1 = 6959^2 ......... (48427680/6) + 1 = 2841^2
274266720 + 1 = 16561^2 ....... (274266720/6) + 1 = 6761^2
4745418768 + 1 = 68887^2 ...... (4745418768/6) + 1 = 28123^2
………………………………………………………..
………………………………………………………..

n = 7

168 + 1 = 13^2 ............... (168/7) + 1 = 5^2
840 + 1 = 29^2 ............... (840/7) + 1 = 11^2
43680 + 1 = 209^2 ............ (43680/7) + 1 = 79^2
214368 + 1 = 463^2 ........... (214368/7) + 1 = 175^2
11095560 + 1 = 3331^2 ........ (11095560/7) + 1 = 1259^2
54449640 + 1 = 7379^2 ........ (54449640/7) + 1 = 2789^2
2818229568 + 1 = 53087^2 ..... (2818229568/7) + 1 = 20065^2
13829995200 + 1 = 117601^2 ... (13829995200/7) + 1 = 44449^2
………………………………………………………..
………………………………………………………..

n = 8

24 + 1 = 5^2 ................ (24/8) + 1 = 2^2 = 1 + T_2
120 + 1 = 11^2 .............. (120/8) + 1 = 4^2 = 1 + T_5
960 + 1 = 31^2 .............. (960/8) + 1 = 11^2 = 1 + T_{15}
4224 + 1 = 65^2 ............. (4224/8) + 1 = 23^2 = 1 + T_{32}
32760 + 1 = 181^2 ........... (32760/8) + 1 = 64^2 = 1 + T_{90}
143640 + 1 = 379^2 .......... (143640/8) + 1 = 134^2 = 1 + T_{189}
1113024 + 1 = 1055^2 ........ (1113024/8) + 1 = 373^2 = 1 + T_{527}
4879680 + 1 = 2209^2 ........ (4879680/8) + 1 = 781^2 = 1 + T_{1104}
37810200 + 1 = 6149^2 ....... (37810200/8) + 1 = 2174^2 = 1 + T_{3074}
165765624 + 1 = 12875^2 ..... (165765624/8) + 1 = 4552^2 = 1 + T_{6437}
……………………………………………………………………….
………………………………………………………………………..

 
 

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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