2024/01/08 by Scott, Reese, Styer, Robert · 1 citation
#11D61 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2401.04197
We show that there are at most two solutions in positive integers (x,y,z) to the equation ax+by=cz for positive integers a, b, and c all greater than one, with just one exceptional case when gcd(a,b)=1, and just one exceptional infinite family of cases when gcd(a,b)>1 (two solutions (x1,y1,z1) and (x2,y2,z2) are considered the same solution if \ ax1, by1 \ = \ ax2, by2 \). The case in which gcd(a,b)=1 has been handled in a series of successive results by Scott and Styer, Hu and Le, and Miyazaki and Pink, who showed that there are at most two solutions, excepting (\a,b\,c) = (\3,5\,2), which gives three solutions. So here we treat the case gcd(a,b)>1, showing that in this case there are at most two solutions, excepting (a,b,c) = (2u, 2v, 2w) with gcd(uv,w)=1, which gives an infinite number of solutions. This generalizes work of Bennett, who proved, for both gcd(a,b)=1 and gcd(a,b)>1, there are at most two solutions (y,z) to the equation a + by = cz, and conjectured there are exactly eleven (a,b,c) giving two solutions to this equation (assuming b and c are not perfect powers). For both gcd(a,b)=1 and gcd(a,b)>1, there are an infinite number of (a,b,c) giving two solutions (x,y,z) to the title equation, which are described in detail in this and a cited previous paper. In a further result, in which we no longer say that two solutions (x1,y1,z1) and (x2,y2,z2) are considered the same solution if \ ax1, by1 \ = \ ax2, by2 \, we list all cases with more than two solutions.