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f(n) is a perfect square for all n

Source: Benelux 2009

January 29, 2011
functionnumber theory proposednumber theory

Problem Statement

Find all functions f:Z>0→Z>0f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0} that satisfy the following two conditions: ∙ f(n)\bullet\ f(n) is a perfect square for all n∈Z>0n\in\mathbb{Z}_{>0} ∙ f(m+n)=f(m)+f(n)+2mn\bullet\ f(m+n)=f(m)+f(n)+2mn for all m,n∈Z>0m,n\in\mathbb{Z}_{>0}.
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