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Number of ways of writing as sum of three integers.

Source: ILL 1979-45

June 2, 2011
modular arithmeticfunctionnumber theory unsolvednumber theory

Problem Statement

For any positive integer nn, we denote by F(n)F(n) the number of ways in which nn can be expressed as the sum of three different positive integers, without regard to order. Thus, since 10=7+2+1=6+3+1=5+4+1=5+3+210 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2, we have F(10)=4F(10) = 4. Show that F(n)F(n) is even if n≡2n \equiv 2 or 4(mod6)4 \pmod 6, but odd if nn is divisible by 66.
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