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Peculiarity of the number 65

 

  1. The number 65 is the "smallest number" which can be expressed as the sum of two different squares in two different ways-
    i.e., 65= 12 +82 = 42+72.
  2. The number 65 is the "smallest number " which can be expressed as the sum of two different cubes of positive integers-
    i.e., 65= 13+43.
  3. The number 65 is the "smallest number" which can be expressed as the sum of squares of two different cubic number as well as the sum of cubes of two different square numbers-
    i.e.,65 = 12+82 where 1 & 8 are cubic numbers.
               = 13+43 where 1 & 4 are square numbers
  4. 4. The number 65 also gives a perfect square on adding or subtracting its reverse from it-
    i.e., 65+56 =121= 112
           65-56 = 9 = 32

 

Moumita Pal
[Epsilon]

 

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