Concatenations Oblong numbers| O(n) || x^2 = A^2

 
 
An Oblong number is a number of the form   n \,(n + 1)

 

O_n \; || \; x^2 \; = \; A^2

Here are the first few solutions for x < 10

 

O_n \; || \; 1^2 \; = \; A^2
 

3(3 + 1) || 1 = 12 || 1 = 121 = 11^2

15(15 + 1) || 1 = 240 || 1 = 2401 = 49^2

132(132 + 1) || 1 = 17556 || 1 = 175561 = 419^2

588(588 + 1) || 1 = 346332 || 1 = 3463321 = 1861^2

5031(5031 + 1) || 1 = 25315992 || 1 = 253159921 = 15911^2

22347(22347 + 1) || 1 = 499410756 || 1 = 4994107561 = 70669^2

191064(191064 + 1) || 1 = 36505643160 || 1 = 365056431601 = 604199^2

848616(848616 + 1) || 1 = 720149964072 || 1 = 7201499640721 = 2683561^2

7255419(7255419 + 1) || 1 = 52641112120980 || 1 = 526411121209801 = 22943651^2

32225079(32225079 + 1) || 1 = 1038455748781320 || 1 = 10384557487813201 = 101904649^2

 
O_n \; || \; 2^2 \; = \; B^2

2(2 + 1) || 2^2 = 6 || 4 = 64 = 8^2

21(21 + 1) || 2^2 = 462 || 4 = 4624 = 68^2

95(95 + 1) || 2^2 = 9120 || 4 = 91204 = 302^2

816(816 + 1) || 2^2 = 666672 || 4 = 6666724 = 2582^2

3626(3626 + 1) || 2^2 = 13151502 || 4 = 131515024 = 11468^2

31005(31005 + 1) || 2^2 = 961341030 || 4 = 9613410304 = 98048^2

137711(137711 + 1) || 2^2 = 18964457232 || 4 = 189644572324 = 435482^2

1177392(1177392 + 1) || 2^2 = 1386253099056 || 4 = 13862530990564 = 3723242^2

5229410(5229410 + 1) || 2^2 = 27346734177510 || 4 = 273467341775104 = 16536848^2

44709909(44709909 + 1) || 2^2 = 1998976007498190 || 4 = 19989760074981904 = 141385148^2

 

O_n \; || \; 3^2 \; = \; C^2

8(8 + 1) || 3^2 = 72 || 9 = 729 = 27^2

27(27 + 1) || 3^2 = 756 || 9 = 7569 = 87^2

323(323 + 1) || 3^2 = 104652 || 9 = 1046529 = 1023^2

1044(1044 + 1) || 3^2 = 1090980 || 9 = 10909809 = 3303^2

12284(12284 + 1) || 3^2 = 150908940 || 9 = 1509089409 = 38847^2

39663(39663 + 1) || 3^2 = 1573193232 || 9 = 15731932329 = 125427^2

466487(466487 + 1) || 3^2 = 217610587656 || 9 = 2176105876569 = 1475163^2

1506168(1506168 + 1) || 3^2 = 2268543550392 || 9 = 22685435503929 = 4762923^2

17714240(17714240 + 1) || 3^2 = 313794316491840 || 9 = 3137943164918409 = 56017347^2

57194739(57194739 + 1) || 3^2 = 3271238226472860 || 9 = 32712382264728609 = 180865647^2

 

O_n \; || \; 5^2 \; = \; D^2
 
   225 = (1\times 2) || 5^2 = 15^2 = (1 || 5)^2
   625 = (2\times 3) || 5^2 = 25^2 = (2 || 5)^2
  1225 = (3\times 4) || 5^2 = 35^2 = (3 || 5)^2
  2025 = (4\times 5) || 5^2 = 45^2 = (4 || 5)^2
  3025 = (5\times 6) || 5^2 = 55^2 = (5 || 5)^2
  4225 = (6\times 7) || 5^2 = 65^2 = (6 || 5)^2
  5625 = (7\times 8) || 5^2 = 75^2 = (7 || 5)^2
  7225 = (8\times 9) || 5^2 = 85^2 = (8 || 5)^2
  9025 = (9\times 10) || 5^2 = 95^2 = (9 || 5)^2
11025 = (10\times 11) || 5^2 = 105^2 = (10 || 5)^2
13225 = (11\times 12) || 5^2 = 115^2 = (11 || 5)^2
15625 = (12\times 13) || 5^2 = 125^2 = (12 || 5)^2
18225 = (13\times 14) || 5^2 = 135^2 = (13 || 5)^2
21025 = (14\times 15) || 5^2 = 145^2 = (14 || 5)^2
24025 = (15\times 16) || 5^2 = 155^2 = (15 || 5)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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