Strongly Nontriangular Numbers

SummaryWe study the question of which numbers are equal to the number of integer points in a right triangle. Let a and b be positive integers and consider the triangle whose corners are (0,0), (a,0), and (0,b). Let P(a,b) denote the number of lattice points in the triangle, excluding boundary points on the hypotenuse. For any b≥2, we say a number n is a b-triangular number if there exists an integer a≥b such that n=P(a,b). We say n is a generalized triangular number if it is b-triangular for some b. If a positive integer is not a generalized triangular number, we say it is strongly nontriangular. We conjecture there are infinitely many strongly nontriangular numbers. We prove several partial results, including a weaker version of this statement. Namely, we show there are infinitely many positive integers that cannot be expressed as P(a,b) for any a,b≥2 with gcd(a,b)=1.