{
  "index": "1954-B-1",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "1. Show that the equation \\( x^{2}-y^{2}=a^{3} \\) has always integral solutions for \\( x \\) and \\( y \\) whenever \\( a \\) is a positive integer.",
  "solution": "Solution. Let \\( x+y=a^{2} \\), and \\( x-y=a \\). Then \\( x^{2}-y^{2}=a^{3} \\), and \\( x=\\frac{1}{2}\\left(a^{2}+a\\right) \\) and \\( y=\\frac{1}{2}\\left(a^{2}-a\\right) \\). Since \\( a^{2} \\) and \\( a \\) are both even or both odd, \\( x \\) and \\( y \\) are both integers and a solution exists for every integer \\( a \\).\n\nRemark. There are other solutions, for example,\n\\[\nx=\\frac{a^{3}+1}{2}, \\quad y=\\frac{a^{3}-1}{2}, \\quad \\text { for } a \\text { odd }\n\\]\nand\n\\[\nx=\\frac{a^{3}+4}{4}, \\quad y=\\frac{a^{3}-4}{4}, \\quad \\text { for } a \\text { even. }\n\\]",
  "vars": [
    "x",
    "y"
  ],
  "params": [
    "a"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "unknownx",
        "y": "unknowny",
        "a": "paramalpha"
      },
      "question": "Problem:\n<<<\n1. Show that the equation \\( unknownx^{2}-unknowny^{2}=paramalpha^{3} \\) has always integral solutions for \\( unknownx \\) and \\( unknowny \\) whenever \\( paramalpha \\) is a positive integer.\n>>>",
      "solution": "Solution:\n<<<\nSolution. Let \\( unknownx+unknowny=paramalpha^{2} \\), and \\( unknownx-unknowny=paramalpha \\). Then \\( unknownx^{2}-unknowny^{2}=paramalpha^{3} \\), and \\( unknownx=\\frac{1}{2}\\left(paramalpha^{2}+paramalpha\\right) \\) and \\( unknowny=\\frac{1}{2}\\left(paramalpha^{2}-paramalpha\\right) \\). Since \\( paramalpha^{2} \\) and \\( paramalpha \\) are both even or both odd, \\( unknownx \\) and \\( unknowny \\) are both integers and a solution exists for every integer \\( paramalpha \\).\n\nRemark. There are other solutions, for example,\n\\[\nunknownx=\\frac{paramalpha^{3}+1}{2}, \\quad unknowny=\\frac{paramalpha^{3}-1}{2}, \\quad \\text { for } paramalpha \\text { odd }\n\\]\nand\n\\[\nunknownx=\\frac{paramalpha^{3}+4}{4}, \\quad unknowny=\\frac{paramalpha^{3}-4}{4}, \\quad \\text { for } paramalpha \\text { even. }\n\\]\n>>>"
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "moonlight",
        "y": "seashells",
        "a": "starlight"
      },
      "question": "1. Show that the equation \\( moonlight^{2}-seashells^{2}=starlight^{3} \\) has always integral solutions for \\( moonlight \\) and \\( seashells \\) whenever \\( starlight \\) is a positive integer.",
      "solution": "Solution. Let \\( moonlight+seashells=starlight^{2} \\), and \\( moonlight-seashells=starlight \\). Then \\( moonlight^{2}-seashells^{2}=starlight^{3} \\), and \\( moonlight=\\frac{1}{2}\\left(starlight^{2}+starlight\\right) \\) and \\( seashells=\\frac{1}{2}\\left(starlight^{2}-starlight\\right) \\). Since \\( starlight^{2} \\) and \\( starlight \\) are both even or both odd, \\( moonlight \\) and \\( seashells \\) are both integers and a solution exists for every integer starlight.\n\nRemark. There are other solutions, for example,\n\\[\nmoonlight=\\frac{starlight^{3}+1}{2}, \\quad seashells=\\frac{starlight^{3}-1}{2}, \\quad \\text { for } starlight \\text { odd }\n\\]\nand\n\\[\nmoonlight=\\frac{starlight^{3}+4}{4}, \\quad seashells=\\frac{starlight^{3}-4}{4}, \\quad \\text { for } starlight \\text { even. }\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantnum",
        "y": "steadydigit",
        "a": "negativeseed"
      },
      "question": "1. Show that the equation \\( constantnum^{2}-steadydigit^{2}=negativeseed^{3} \\) has always integral solutions for \\( constantnum \\) and \\( steadydigit \\) whenever \\( negativeseed \\) is a positive integer.",
      "solution": "Solution. Let \\( constantnum+steadydigit=negativeseed^{2} \\), and \\( constantnum-steadydigit=negativeseed \\). Then \\( constantnum^{2}-steadydigit^{2}=negativeseed^{3} \\), and \\( constantnum=\\frac{1}{2}\\left(negativeseed^{2}+negativeseed\\right) \\) and \\( steadydigit=\\frac{1}{2}\\left(negativeseed^{2}-negativeseed\\right) \\). Since \\( negativeseed^{2} \\) and \\( negativeseed \\) are both even or both odd, \\( constantnum \\) and \\( steadydigit \\) are both integers and a solution exists for every integer \\( negativeseed \\).\n\nRemark. There are other solutions, for example,\n\\[\nconstantnum=\\frac{negativeseed^{3}+1}{2}, \\quad steadydigit=\\frac{negativeseed^{3}-1}{2}, \\quad \\text { for } negativeseed \\text { odd }\n\\]\nand\n\\[\nconstantnum=\\frac{negativeseed^{3}+4}{4}, \\quad steadydigit=\\frac{negativeseed^{3}-4}{4}, \\quad \\text { for } negativeseed \\text { even. }\n\\]"
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "a": "mnlkprqs"
      },
      "question": "1. Show that the equation \\( qzxwvtnp^{2}-hjgrksla^{2}=mnlkprqs^{3} \\) has always integral solutions for \\( qzxwvtnp \\) and \\( hjgrksla \\) whenever \\( mnlkprqs \\) is a positive integer.",
      "solution": "Solution. Let \\( qzxwvtnp+hjgrksla=mnlkprqs^{2} \\), and \\( qzxwvtnp-hjgrksla=mnlkprqs \\). Then \\( qzxwvtnp^{2}-hjgrksla^{2}=mnlkprqs^{3} \\), and \\( qzxwvtnp=\\frac{1}{2}\\left(mnlkprqs^{2}+mnlkprqs\\right) \\) and \\( hjgrksla=\\frac{1}{2}\\left(mnlkprqs^{2}-mnlkprqs\\right) \\). Since \\( mnlkprqs^{2} \\) and \\( mnlkprqs \\) are both even or both odd, \\( qzxwvtnp \\) and \\( hjgrksla \\) are both integers and a solution exists for every integer \\( mnlkprqs \\).\n\nRemark. There are other solutions, for example,\n\\[\nqzxwvtnp=\\frac{mnlkprqs^{3}+1}{2}, \\quad hjgrksla=\\frac{mnlkprqs^{3}-1}{2}, \\quad \\text { for } mnlkprqs \\text { odd }\n\\]\nand\n\\[\nqzxwvtnp=\\frac{mnlkprqs^{3}+4}{4}, \\quad hjgrksla=\\frac{mnlkprqs^{3}-4}{4}, \\quad \\text { for } mnlkprqs \\text { even. }\n\\]"
    },
    "kernel_variant": {
      "question": "Let a be any non-zero integer.  Prove that the Diophantine equation  \n  x² − y² = a⁶  \npossesses an integral solution (x, y) that also satisfies  \n  gcd(x, y) = a.",
      "solution": "Put x = a u, y = a v; then u² − v² = a⁴.  For a odd choose u = (a⁴+1)/2, v = (a⁴−1)/2; for a even write a = 2k and set u = (a⁴+4)/4, v = (a⁴−4)/4.  These choices are integral.  Note that extracting the common factor a at the outset lets us govern gcd(x, y) conveniently.  \n\nNow u² − v² = (u+v)(u−v) = a⁴, so a²(u² − v²) = a⁶, giving x² − y² = a⁶.  Moreover u − v equals 1 or 2, hence any common divisor of u and v divides that difference, so gcd(u, v) = 1 and therefore gcd(x, y) = a.  The construction works for every non-zero integer a, completing the proof.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.110652",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}