Diophantine representation of perfect numbers

There is explicitly given a polynomial (of degree 17 in 48 variables), the set of whose positive values (for natural values of the variables) is precisely the set of all perfect numbers. The construction of this polynomial is based on a new test for a number to be perfect, formulated in terms of the divisibility of binomial coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 78–89, 1979.