11 Feb '03 22:08

...in under a week.

Take one dot. From there, you can make a 3-dot equilateral triangle, a four-dot square, a five dot regular pentagon, etc. From those, you can build a 6-dot trianlge, a 9-dot square, a 12-dot pentagon, etc. So any number that is the total number of dots in a regular r-gon so built is a r-gonal number. For convenience, say 0 is an r-gonal number for any r. Can someone prove that EVERY positive whole number can be represented in at least one way as the sum of r r-gonal numbers, for every r?

Hints as needed, although I bet someone can come up with a prettier proof than mine...

Take one dot. From there, you can make a 3-dot equilateral triangle, a four-dot square, a five dot regular pentagon, etc. From those, you can build a 6-dot trianlge, a 9-dot square, a 12-dot pentagon, etc. So any number that is the total number of dots in a regular r-gon so built is a r-gonal number. For convenience, say 0 is an r-gonal number for any r. Can someone prove that EVERY positive whole number can be represented in at least one way as the sum of r r-gonal numbers, for every r?

Hints as needed, although I bet someone can come up with a prettier proof than mine...