(x + y)(x + y + 1) = k*x*y

 
 
(x + y) \,(x + y + 1) \; = \; k \, x \, y

An interesting case of when    k \; = \; 4

    81\times 80 \; = \; 4\times 45\times 36 \; = \; 6480
   289\times 288 \; = \; 4\times 153\times 136 \; = \; 83232
   625\times 624 \; = \; 4\times 325\times 300 \; = \; 390000
  1089\times 1088 \; = \; 4\times 561\times 528 \; = \; 1184832
  1681\times 1680 \; = \; 4\times 861\times 820 \; = \; 2824080
  2401\times 2400 \; = \; 4\times 1225\times 1176 \; = \; 5762400
  3249\times 3248 \; = \; 4\times 1653\times 1596 \; = \; 10552752
  4225\times 4224 \; = \; 4\times 2145\times 2080 \; = \; 17846400
  5329\times 5328 \; = \; 4\times 2701\times 2628 \; = \; 28392912
  6561\times 6560 \; = \; 4\times 3321\times 3240 \; = \; 43040160
  7921\times 7920 \; = \; 4\times 4005\times 3916 \; = \; 62734320
  9409\times 9408 \; = \; 4\times 4753\times 4656 \; = \; 88519872
11025\times 11024 \; = \; 4\times 5565\times 5460 \; = \; 121539600
12769\times 12768 \; = \; 4\times 6441\times 6328 \; = \; 163034592
14641\times 14640 \; = \; 4\times 7381\times 7260 \; = \; 214344240

81,   289,   625,   1089,   …   are of the form   (8 \, n + 1)^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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