On partitions into four distinct squares of equal parity (Q2760450)

In a previous paper [Discrete Math. 211, No. 1-3, 225-228 (2000; Zbl 0941.11039)] the author discussed the number of partitions of \(n\) into four squares. He now studies \(p_{4e}(n)\), \(p_{4o}(n)\) and \(p_{4e+}(n)\), the numbers of partitions of \(n\) into the sum of four distinct even squares, odd squares and positive even squares, respectively. He shows that if \(n\equiv 4\pmod 8\) then \(p_{4o}(n)= p_{4e}(n)+ p_{4e+}(n)\), from which it follows that if \(p_{4o}(n)> 0\) then \(p_{4e}(n)> 0\), a result conjectured in 1987 by R. W. Gosper.