NESTED SQUARES AND EVALUATIONS OF INTEGER PRODUCTS

KARL DILCHER

Abstract. The identity

((x^{2} − 85)^{2} − 4176)^{2} − 2880^{2} = (x^{2} − 1^{2})(x^{2} − 7^{2})(x^{2} − 11^{2})(x^{2} − 13^{2}),

discovered by R. E. Crandall, allows the evaluation of a product of 8 integers by a succession of 3 squares and 3 subtractions. The question arises whether there exist formulas like Crandall’s with more than 3 nested squares. It will be shown that this is not the case; however, there are infinitely many formulas of length 3.