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partitions

Let f(k) be the sum of the first k odd natural numbers; that is, f(k)=1+2++(2k?1).  Compute the values f(k) for several values.  Can you see a pattern?  Can you prove the pattern?
Which positive integers can be written as the sum of two squares of (nonnegative) integers?  For example, 4=02+22, 5=12+22, and 8=22+22.  However, 3 cannot be written as the sum of two squares.  Determine which of the first 100 natural numbers can be written as the sum of two squares.  What patterns do you observe? How could you gain more evidence to support your conjectures?

 2 retagged FrédéricC 3139 ●3 ●33 ●61

partitions

Let f(k) be the sum of the first k odd natural numbers; that is, f(k)=1+2++(2k?1).  Compute the values f(k) for several values.  Can you see a pattern?  Can you prove the pattern?
Which positive integers can be written as the sum of two squares of (nonnegative) integers?  For example, 4=02+22, 5=12+22, and 8=22+22.  However, 3 cannot be written as the sum of two squares.  Determine which of the first 100 natural numbers can be written as the sum of two squares.  What patterns do you observe? How could you gain more evidence to support your conjectures?