13=1

13+23=9  = 32                         

13+23+33=36 = 62

13+23+33+43=100 = 102

13+23+33+43+53=225 = 152

…. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16….

These are triangular numbers are in bold: 

  • 1+2 =3
  • 1+2+3=6
  • 1+2+3+4=10

          ( n(n+1)/2 )2= n2(n+1)2/4

A question I have in my mind is that Fermat’s last theorem states: If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c

But how about  a3 + b3 + c3 = d       … Are there any integer solutions to this?   I ask this because geometrically speaking volume is 1 degree of freedom more than area.

      33+43+53=63             …. = 152 – 32

 

 

Categories: Math

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