Lagrange’s theorem tells us that the quadratic form x2 + y2 + z2 + t2 represents all positive integers. What about the more general form ax2 + by2 + cz2 + dt2, where a, b, c, and d are positive integers? We provide a gentle introduction to this question by posing and answering some natural questions about the possible integers that ax2 + by2 + cz2 + dt2 can represent.

MSC: Primary 11E20
American Mathematical Monthly
School of Mathematics and Statistics

Williams, K.S. (2018). Everything You Wanted To Know About ax2+by2+cz2+dt2 But Were Afraid To Ask. American Mathematical Monthly, 125(9), 797–810. doi:10.1080/00029890.2018.1503003