{
  "equation": "43m\u221249n=13",
  "source_solution": [
    6,
    5
  ],
  "all_integer_solutions": "m=6+49k,n=5+43k for integerk",
  "proof": "Subtract theknownsolution:43(m\u22126)=49(n\u22125);coprimality of43and49 forces the stated steps.",
  "positive_solutions": "k>=0; the source solution is the smallest positive pair in this fixed coefficientequation.",
  "examples": [
    [
      6,
      5
    ],
    [
      55,
      48
    ]
  ],
  "limitation": "The equationdoesnot identifyMoses orauthorizeotherchronologicalendpoints; fullsourcepath2580 supplies the application."
}