{
  "index": "1959-A-1",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "1. Let \\( n \\) be a positive integer. Prove that \\( x^{n}-\\left(1 / x^{n}\\right) \\) is expressible as a polynomial in \\( x-(1 / x) \\) with real coefficients if and only if \\( n \\) is odd.",
  "solution": "Solution. Let \\( z=x-1 / x \\). If the desired representation\n\\[\nx^{\\prime \\prime}-\\left(\\frac{1}{x}\\right)^{\\prime \\prime}=P_{n}(z)\n\\]\nexists, the coefficient of \\( z^{\\prime \\prime} \\) in \\( P_{\"} \\) must be one. Then equating the terms in \\( 1 / x^{\\prime \\prime} \\), we see that \\( -1 / x^{\\prime \\prime}=(-1 / x)^{\\prime \\prime} \\), which implies that \\( n \\) is odd.\n\nTo show conversely that the representations exist for all odd \\( n \\), we use induction. Clearly they exist for \\( n=1,3 \\) with \\( P_{1}(z)=z, P_{3}(z)=z^{3}+3 z \\). Suppose representations exist for \\( n=1,3, \\ldots, 2 k-1 \\), where \\( k \\geq 2 \\). Then\n\\[\n\\left(x^{2}+\\frac{1}{x^{2}}\\right)\\left(x^{2 k-1}-\\frac{1}{x^{2 k-1}}\\right)=x^{2 k+1}-\\frac{1}{x^{2 k+1}}+x^{2 k} 3-\\frac{1}{x^{2 k-3}}\n\\]\nthat is,\n\\[\n\\left(z^{2}+2\\right) P_{2 k} \\quad(z)=x^{2 k+1}-1 / x^{2 k+1}+P_{2 k \\cdot 3}(z)\n\\]\n\nSo we define\n\\[\nP_{2 k+1}(y)=\\left(y^{2}+2\\right) P_{2 k} 1(y)-P_{2 k-3}(y)\n\\]\nand we have\n\\[\nP_{2 k+1}(z)=x^{2 k+1}-1 / x^{2 k+1}\n\\]\nand there is such a polynomial for \\( n=2 k+1 \\). Therefore \\( x^{n}-1 / x^{n} \\) can be written as a polynomial in \\( x-1 / x \\) with integer coefficients for all odd positive integers \\( n \\).\n\nRemarks. For \\( n \\) a positive even integer, say \\( n=2 k \\), there exists no function \\( Q \\). polynomial or otherwise, such that\n\\[\nx^{n}-\\frac{1}{x^{n}}=Q\\left(x-\\frac{1}{x}\\right)\n\\]\n\nIf such a function did exist, by putting \\( x=\\frac{1}{2} \\) and \\( x=-2 \\) successively we would have\n\\[\n\\frac{1}{2^{2 k}}-2^{2 k}=Q\\left(-\\frac{3}{2}\\right)=2^{2 k}-\\frac{1}{2^{2 k}},\n\\]\na contradiction.\nIt is easy to see that \\( x^{n}+1 / x^{n} \\) can be represented as a polynomial in \\( x+1 / x \\) for all positive integers \\( n \\). Putting \\( x=e^{i \\theta} \\), this shows that \\( \\cos n \\theta \\) is a polynomial in \\( \\cos \\theta \\) for all \\( n \\). The same substitution in the result of the problem shows that \\( \\sin n \\theta \\) is a polynomial in \\( \\sin \\theta \\) for odd \\( n \\); but, of course, it is not for even \\( n \\).",
  "vars": [
    "n",
    "x",
    "z",
    "k",
    "Q",
    "P_n",
    "P_1",
    "P_3",
    "P_2k-1",
    "P_2k",
    "P_2k+1",
    "P_2k-3",
    "y",
    "\\\\theta"
  ],
  "params": [],
  "sci_consts": [
    "e",
    "i"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "powerindex",
        "x": "corevariable",
        "z": "auxiliary",
        "k": "halfindex",
        "Q": "targetfunc",
        "P_n": "polyfamily",
        "P_1": "polyone",
        "P_3": "polythree",
        "P_2k-1": "polytwokminusone",
        "P_2k": "polytwok",
        "P_2k+1": "polytwokplusone",
        "P_2k-3": "polytwokminusthree",
        "y": "tempvar",
        "\\theta": "angletheta"
      },
      "question": "1. Let \\( powerindex \\) be a positive integer. Prove that \\( corevariable^{powerindex}-\\left(1 / corevariable^{powerindex}\\right) \\) is expressible as a polynomial in \\( corevariable-(1 / corevariable) \\) with real coefficients if and only if \\( powerindex \\) is odd.",
      "solution": "Solution. Let \\( auxiliary=corevariable-1 / corevariable \\). If the desired representation\n\\[\ncorevariable^{\\prime \\prime}-\\left(\\frac{1}{corevariable}\\right)^{\\prime \\prime}=polyfamily(auxiliary)\n\\]\nexists, the coefficient of \\( auxiliary^{\\prime \\prime} \\) in \\( polyfamily \\) must be one. Then equating the terms in \\( 1 / corevariable^{\\prime \\prime} \\), we see that \\( -1 / corevariable^{\\prime \\prime}=(-1 / corevariable)^{\\prime \\prime} \\), which implies that \\( powerindex \\) is odd.\n\nTo show conversely that the representations exist for all odd \\( powerindex \\), we use induction. Clearly they exist for \\( powerindex=1,3 \\) with \\( polyone(auxiliary)=auxiliary, polythree(auxiliary)=auxiliary^{3}+3 auxiliary \\). Suppose representations exist for \\( powerindex=1,3, \\ldots, 2 halfindex-1 \\), where \\( halfindex \\geq 2 \\). Then\n\\[\n\\left(corevariable^{2}+\\frac{1}{corevariable^{2}}\\right)\\left(corevariable^{2 halfindex-1}-\\frac{1}{corevariable^{2 halfindex-1}}\\right)=corevariable^{2 halfindex+1}-\\frac{1}{corevariable^{2 halfindex+1}}+corevariable^{2 halfindex} 3-\\frac{1}{corevariable^{2 halfindex-3}}\n\\]\nthat is,\n\\[\n\\left(auxiliary^{2}+2\\right) polytwok \\quad(auxiliary)=corevariable^{2 halfindex+1}-1 / corevariable^{2 halfindex+1}+polytwokminusthree(auxiliary)\n\\]\n\nSo we define\n\\[\npolytwokplusone(tempvar)=\\left(tempvar^{2}+2\\right) polytwok 1(tempvar)-polytwokminusthree(tempvar)\n\\]\nand we have\n\\[\npolytwokplusone(auxiliary)=corevariable^{2 halfindex+1}-1 / corevariable^{2 halfindex+1}\n\\]\nand there is such a polynomial for \\( powerindex=2 halfindex+1 \\). Therefore \\( corevariable^{powerindex}-1 / corevariable^{powerindex} \\) can be written as a polynomial in \\( corevariable-1 / corevariable \\) with integer coefficients for all odd positive integers \\( powerindex \\).\n\nRemarks. For \\( powerindex \\) a positive even integer, say \\( powerindex=2 halfindex \\), there exists no function \\( targetfunc \\), polynomial or otherwise, such that\n\\[\ncorevariable^{powerindex}-\\frac{1}{corevariable^{powerindex}}=targetfunc\\left(corevariable-\\frac{1}{corevariable}\\right)\n\\]\n\nIf such a function did exist, by putting \\( corevariable=\\frac{1}{2} \\) and \\( corevariable=-2 \\) successively we would have\n\\[\n\\frac{1}{2^{2 halfindex}}-2^{2 halfindex}=targetfunc\\left(-\\frac{3}{2}\\right)=2^{2 halfindex}-\\frac{1}{2^{2 halfindex}},\n\\]\na contradiction.\nIt is easy to see that \\( corevariable^{powerindex}+1 / corevariable^{powerindex} \\) can be represented as a polynomial in \\( corevariable+1 / corevariable \\) for all positive integers \\( powerindex \\). Putting \\( corevariable=e^{i angletheta} \\), this shows that \\( \\cos powerindex angletheta \\) is a polynomial in \\( \\cos angletheta \\) for all \\( powerindex \\). The same substitution in the result of the problem shows that \\( \\sin powerindex angletheta \\) is a polynomial in \\( \\sin angletheta \\) for odd \\( powerindex \\); but, of course, it is not for even \\( powerindex \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "sandcastle",
        "x": "pinebranch",
        "z": "moonshadow",
        "k": "riverstone",
        "Q": "lanternfish",
        "P_n": "winterberry",
        "P_1": "driftwood",
        "P_3": "starlizard",
        "P_2k-1": "seasparrow",
        "P_2k": "cloudanchor",
        "P_2k+1": "fogharbor",
        "P_2k-3": "dunecompass",
        "y": "thornapple",
        "\\\\theta": "cinderflare"
      },
      "question": "1. Let \\( sandcastle \\) be a positive integer. Prove that \\( pinebranch^{sandcastle}-\\left(1 / pinebranch^{sandcastle}\\right) \\) is expressible as a polynomial in \\( pinebranch-(1 / pinebranch) \\) with real coefficients if and only if \\( sandcastle \\) is odd.",
      "solution": "Solution. Let \\( moonshadow=pinebranch-1 / pinebranch \\). If the desired representation\n\\[\npinebranch^{\\prime \\prime}-\\left(\\frac{1}{pinebranch}\\right)^{\\prime \\prime}=winterberry(moonshadow)\n\\]\nexists, the coefficient of \\( moonshadow^{\\prime \\prime} \\) in \\( winterberry \\) must be one. Then equating the terms in \\( 1 / pinebranch^{\\prime \\prime} \\), we see that \\( -1 / pinebranch^{\\prime \\prime}=(-1 / pinebranch)^{\\prime \\prime} \\), which implies that \\( sandcastle \\) is odd.\n\nTo show conversely that the representations exist for all odd \\( sandcastle \\), we use induction. Clearly they exist for \\( sandcastle=1,3 \\) with \\( driftwood(moonshadow)=moonshadow, starlizard(moonshadow)=moonshadow^{3}+3 moonshadow \\). Suppose representations exist for \\( sandcastle=1,3, \\ldots, 2 riverstone-1 \\), where \\( riverstone \\geq 2 \\). Then\n\\[\n\\left(pinebranch^{2}+\\frac{1}{pinebranch^{2}}\\right)\\left(pinebranch^{2 riverstone-1}-\\frac{1}{pinebranch^{2 riverstone-1}}\\right)=pinebranch^{2 riverstone+1}-\\frac{1}{pinebranch^{2 riverstone+1}}+pinebranch^{2 riverstone} 3-\\frac{1}{pinebranch^{2 riverstone-3}}\n\\]\nthat is,\n\\[\n\\left(moonshadow^{2}+2\\right) cloudanchor \\quad(moonshadow)=pinebranch^{2 riverstone+1}-1 / pinebranch^{2 riverstone+1}+dunecompass(moonshadow)\n\\]\n\nSo we define\n\\[\nfogharbor(thornapple)=\\left(thornapple^{2}+2\\right) cloudanchor 1(thornapple)-dunecompass(thornapple)\n\\]\nand we have\n\\[\nfogharbor(moonshadow)=pinebranch^{2 riverstone+1}-1 / pinebranch^{2 riverstone+1}\n\\]\nand there is such a polynomial for \\( sandcastle=2 riverstone+1 \\). Therefore \\( pinebranch^{sandcastle}-1 / pinebranch^{sandcastle} \\) can be written as a polynomial in \\( pinebranch-1 / pinebranch \\) with integer coefficients for all odd positive integers \\( sandcastle \\).\n\nRemarks. For \\( sandcastle \\) a positive even integer, say \\( sandcastle=2 riverstone \\), there exists no function \\( lanternfish \\). polynomial or otherwise, such that\n\\[\npinebranch^{sandcastle}-\\frac{1}{pinebranch^{sandcastle}}=lanternfish\\left(pinebranch-\\frac{1}{pinebranch}\\right)\n\\]\n\nIf such a function did exist, by putting \\( pinebranch=\\frac{1}{2} \\) and \\( pinebranch=-2 \\) successively we would have\n\\[\n\\frac{1}{2^{2 riverstone}}-2^{2 riverstone}=lanternfish\\left(-\\frac{3}{2}\\right)=2^{2 riverstone}-\\frac{1}{2^{2 riverstone}},\n\\]\na contradiction.\nIt is easy to see that \\( pinebranch^{sandcastle}+1 / pinebranch^{sandcastle} \\) can be represented as a polynomial in \\( pinebranch+1 / pinebranch \\) for all positive integers \\( sandcastle \\). Putting \\( pinebranch=e^{i cinderflare} \\), this shows that \\( \\cos sandcastle cinderflare \\) is a polynomial in \\( \\cos cinderflare \\) for all \\( sandcastle \\). The same substitution in the result of the problem shows that \\( \\sin sandcastle cinderflare \\) is a polynomial in \\( \\sin cinderflare \\) for odd \\( sandcastle \\); but, of course, it is not for even \\( sandcastle \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "zerolength",
        "x": "steadfast",
        "z": "unchanged",
        "k": "finishline",
        "Q": "answerless",
        "P_n": "irrational",
        "P_1": "intangible",
        "P_3": "invisible",
        "P_2k-1": "inconsistent",
        "P_2k": "impermanent",
        "P_2k+1": "undefined",
        "P_2k-3": "ambiguous",
        "y": "inputless",
        "\\\\theta": "straightline"
      },
      "question": "1. Let \\( zerolength \\) be a positive integer. Prove that \\( steadfast^{zerolength}-\\left(1 / steadfast^{zerolength}\\right) \\) is expressible as a polynomial in \\( steadfast-(1 / steadfast) \\) with real coefficients if and only if \\( zerolength \\) is odd.",
      "solution": "Solution. Let \\( unchanged=steadfast-1 / steadfast \\). If the desired representation\n\\[\nsteadfast^{\\prime \\prime}-\\left(\\frac{1}{steadfast}\\right)^{\\prime \\prime}=irrational(unchanged)\n\\]\nexists, the coefficient of \\( unchanged^{\\prime \\prime} \\) in \\( irrational \\) must be one. Then equating the terms in \\( 1 / steadfast^{\\prime \\prime} \\), we see that \\( -1 / steadfast^{\\prime \\prime}=(-1 / steadfast)^{\\prime \\prime} \\), which implies that \\( zerolength \\) is odd.\n\nTo show conversely that the representations exist for all odd \\( zerolength \\), we use induction. Clearly they exist for \\( zerolength=1,3 \\) with \\( intangible(unchanged)=unchanged, invisible(unchanged)=unchanged^{3}+3\\,unchanged \\). Suppose representations exist for \\( zerolength=1,3, \\ldots, 2 finishline-1 \\), where \\( finishline \\geq 2 \\). Then\n\\[\n\\left(steadfast^{2}+\\frac{1}{steadfast^{2}}\\right)\\left(steadfast^{2 finishline-1}-\\frac{1}{steadfast^{2 finishline-1}}\\right)=steadfast^{2 finishline+1}-\\frac{1}{steadfast^{2 finishline+1}}+steadfast^{2 finishline} 3-\\frac{1}{steadfast^{2 finishline-3}}\n\\]\nthat is,\n\\[\n\\left(unchanged^{2}+2\\right) impermanent \\quad(unchanged)=steadfast^{2 finishline+1}-1 / steadfast^{2 finishline+1}+ambiguous(unchanged)\n\\]\n\nSo we define\n\\[\nundefined(inputless)=\\left(inputless^{2}+2\\right) impermanent 1(inputless)-ambiguous(inputless)\n\\]\nand we have\n\\[\nundefined(unchanged)=steadfast^{2 finishline+1}-1 / steadfast^{2 finishline+1}\n\\]\nand there is such a polynomial for \\( zerolength=2 finishline+1 \\). Therefore \\( steadfast^{zerolength}-1 / steadfast^{zerolength} \\) can be written as a polynomial in \\( steadfast-1 / steadfast \\) with integer coefficients for all odd positive integers \\( zerolength \\).\n\nRemarks. For \\( zerolength \\) a positive even integer, say \\( zerolength=2 finishline \\), there exists no function \\( answerless \\). polynomial or otherwise, such that\n\\[\nsteadfast^{zerolength}-\\frac{1}{steadfast^{zerolength}}=answerless\\left(steadfast-\\frac{1}{steadfast}\\right)\n\\]\n\nIf such a function did exist, by putting \\( steadfast=\\frac{1}{2} \\) and \\( steadfast=-2 \\) successively we would have\n\\[\n\\frac{1}{2^{2 finishline}}-2^{2 finishline}=answerless\\left(-\\frac{3}{2}\\right)=2^{2 finishline}-\\frac{1}{2^{2 finishline}},\n\\]\na contradiction.\nIt is easy to see that \\( steadfast^{zerolength}+1 / steadfast^{zerolength} \\) can be represented as a polynomial in \\( steadfast+1 / steadfast \\) for all positive integers \\( zerolength \\). Putting \\( steadfast=e^{i\\, straightline} \\), this shows that \\( \\cos zerolength\\,straightline \\) is a polynomial in \\( \\cos straightline \\) for all \\( zerolength \\). The same substitution in the result of the problem shows that \\( \\sin zerolength\\,straightline \\) is a polynomial in \\( \\sin straightline \\) for odd \\( zerolength \\); but, of course, it is not for even \\( zerolength \\)."
    },
    "garbled_string": {
      "map": {
        "n": "knmdpsej",
        "x": "flgtwzoh",
        "z": "mvrplqda",
        "k": "vxrbnchu",
        "Q": "qzjfrldx",
        "P_n": "qzxwvtnp",
        "P_1": "hjgrksla",
        "P_3": "bmtcfqae",
        "P_2k-1": "skdjphre",
        "P_2k": "uvljrpso",
        "P_2k+1": "wavbncie",
        "P_2k-3": "yrhqlkfd",
        "y": "dfkngpwm",
        "\\\\theta": "gmvxqazo"
      },
      "question": "1. Let \\( knmdpsej \\) be a positive integer. Prove that \\( flgtwzoh^{knmdpsej}-\\left(1 / flgtwzoh^{knmdpsej}\\right) \\) is expressible as a polynomial in \\( flgtwzoh-(1 / flgtwzoh) \\) with real coefficients if and only if \\( knmdpsej \\) is odd.",
      "solution": "Solution. Let \\( mvrplqda=flgtwzoh-1 / flgtwzoh \\). If the desired representation\n\\[\nflgtwzoh^{\\prime \\prime}-\\left(\\frac{1}{flgtwzoh}\\right)^{\\prime \\prime}=qzxwvtnp(mvrplqda)\n\\]\nexists, the coefficient of \\( mvrplqda^{\\prime \\prime} \\) in \\( qzxwvtnp \\) must be one. Then equating the terms in \\( 1 / flgtwzoh^{\\prime \\prime} \\), we see that \\( -1 / flgtwzoh^{\\prime \\prime}=(-1 / flgtwzoh)^{\\prime \\prime} \\), which implies that \\( knmdpsej \\) is odd.\n\nTo show conversely that the representations exist for all odd \\( knmdpsej \\), we use induction. Clearly they exist for \\( knmdpsej=1,3 \\) with \\( hjgrksla(mvrplqda)=mvrplqda, bmtcfqae(mvrplqda)=mvrplqda^{3}+3 mvrplqda \\). Suppose representations exist for \\( knmdpsej=1,3, \\ldots, 2 vxrbnchu-1 \\), where \\( vxrbnchu \\geq 2 \\). Then\n\\[\n\\left(flgtwzoh^{2}+\\frac{1}{flgtwzoh^{2}}\\right)\\left(flgtwzoh^{2 vxrbnchu-1}-\\frac{1}{flgtwzoh^{2 vxrbnchu-1}}\\right)=flgtwzoh^{2 vxrbnchu+1}-\\frac{1}{flgtwzoh^{2 vxrbnchu+1}}+flgtwzoh^{2 vxrbnchu} 3-\\frac{1}{flgtwzoh^{2 vxrbnchu-3}}\n\\]\nthat is,\n\\[\n\\left(mvrplqda^{2}+2\\right) uvljrpso \\quad(mvrplqda)=flgtwzoh^{2 vxrbnchu+1}-1 / flgtwzoh^{2 vxrbnchu+1}+yrhqlkfd(mvrplqda)\n\\]\n\nSo we define\n\\[\nwavbncie(dfkngpwm)=\\left(dfkngpwm^{2}+2\\right) uvljrpso 1(dfkngpwm)-yrhqlkfd(dfkngpwm)\n\\]\nand we have\n\\[\nwavbncie(mvrplqda)=flgtwzoh^{2 vxrbnchu+1}-1 / flgtwzoh^{2 vxrbnchu+1}\n\\]\nand there is such a polynomial for \\( knmdpsej=2 vxrbnchu+1 \\). Therefore \\( flgtwzoh^{knmdpsej}-1 / flgtwzoh^{knmdpsej} \\) can be written as a polynomial in \\( flgtwzoh-1 / flgtwzoh \\) with integer coefficients for all odd positive integers \\( knmdpsej \\).\n\nRemarks. For \\( knmdpsej \\) a positive even integer, say \\( knmdpsej=2 vxrbnchu \\), there exists no function \\( qzjfrldx \\). polynomial or otherwise, such that\n\\[\nflgtwzoh^{knmdpsej}-\\frac{1}{flgtwzoh^{knmdpsej}}=qzjfrldx\\left(flgtwzoh-\\frac{1}{flgtwzoh}\\right)\n\\]\n\nIf such a function did exist, by putting \\( flgtwzoh=\\frac{1}{2} \\) and \\( flgtwzoh=-2 \\) successively we would have\n\\[\n\\frac{1}{2^{2 vxrbnchu}}-2^{2 vxrbnchu}=qzjfrldx\\left(-\\frac{3}{2}\\right)=2^{2 vxrbnchu}-\\frac{1}{2^{2 vxrbnchu}},\n\\]\na contradiction.\nIt is easy to see that \\( flgtwzoh^{knmdpsej}+1 / flgtwzoh^{knmdpsej} \\) can be represented as a polynomial in \\( flgtwzoh+1 / flgtwzoh \\) for all positive integers \\( knmdpsej \\). Putting \\( flgtwzoh=e^{i gmvxqazo} \\), this shows that \\( \\cos knmdpsej gmvxqazo \\) is a polynomial in \\( \\cos gmvxqazo \\) for all \\( knmdpsej \\). The same substitution in the result of the problem shows that \\( \\sin knmdpsej gmvxqazo \\) is a polynomial in \\( \\sin gmvxqazo \\) for odd \\( knmdpsej \\); but, of course, it is not for even \\( knmdpsej \\)."
    },
    "kernel_variant": {
      "question": "Let n be a positive integer.  Prove that the expression\n\\[\n      x^{n}-x^{-n}\n\\]\ncan be written as a polynomial with \\emph{rational} coefficients in the single variable\n\\[\n      x-\\,x^{-1}\n\\]\nif and only if n is odd.",
      "solution": "Set z = x - x^{-1}.  We show first that no such polynomial exists when n is even, and then construct one for every odd n.\n\n1. The obstruction for even n.\n   Suppose that for some even n we had\n        x^{n}-x^{-n}=P_n(z)\n   with P_n∈ℚ[z].  Choose the two numbers\n        x_1 = 2,\n        x_2 = -\\tfrac12.\n   They satisfy\n        z_1 = x_1 - x_1^{-1} = 2 - \\tfrac12 = \\tfrac32,\n        z_2 = x_2 - x_2^{-1} = -\\tfrac12 - (-2) = \\tfrac32,\n   so z_1 = z_2.  But for even n,\n        x_1^{n} - x_1^{-n} = 2^{n} - 2^{-n} > 0,\n        x_2^{n} - x_2^{-n} = (-\\tfrac12)^{n} - (-\\tfrac12)^{-n} = 2^{-n} - 2^{n} < 0.\n   Since P_n(z_1)=P_n(z_2) would imply 2^{n}-2^{-n} = -(2^{n}-2^{-n}), a contradiction, no such P_n exists when n is even.\n\n2. Base cases for odd n.\n   For n=1 and n=3 one checks directly\n        x - x^{-1} = z,\n        x^{3} - x^{-3} = z^{3} + 3z,\n   so we may take P_1(z)=z and P_3(z)=z^{3}+3z, both in ℚ[z].\n\n3. A recurrence relation.\n   The identity\n        (x^{2} + x^{-2})(x^{m} - x^{-m}) = x^{m+2} - x^{-(m+2)} + x^{m-2} - x^{-(m-2)}\n   holds for every integer m.  Since x^{2}+x^{-2} = (x-x^{-1})^{2} + 2 = z^{2}+2, we get the polynomial recurrence\n        P_{m+2}(z) = (z^{2}+2)\\,P_{m}(z) - P_{m-2}(z).\n\n4. Inductive construction for all odd n.\n   Assume for some odd m≥3 that P_{m}(z) and P_{m-2}(z) in ℚ[z] satisfy x^{k}-x^{-k}=P_{k}(z) for k=m and k=m-2.  Substituting into the recurrence yields\n        P_{m+2}(z) = x^{m+2}-x^{-(m+2)},\n   so P_{m+2} has rational coefficients.  Since m odd implies m+2 odd, starting from the base cases n=1,3 and iterating gives a valid P_n for every odd n.\n\n5. Conclusion.\n   There is no representation for even n, and there is one for every odd n.  Hence x^{n}-x^{-n} is a polynomial in x-x^{-1} over ℚ if and only if n is odd.",
      "_meta": {
        "core_steps": [
          "Assume x^n - x^{-n} = P(z) with z = x - x^{-1}; pick two x with identical z but opposite target sign to show even n impossible.",
          "Establish base cases for odd n: P_1(z)=z and P_3(z)=z^3+3z.",
          "Use identity (x^2 + x^{-2})(x^m - x^{-m}) = (x^{m+2} - x^{-(m+2)}) + (x^{m-2} - x^{-(m-2)}) to obtain recurrence P_{m+2}(z) = (z^2+2)P_m(z) - P_{m-2}(z).",
          "Apply induction on odd n via the recurrence to construct P_n for all odd n.",
          "Conclude: representation exists iff n is odd."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Specific pair of x-values that give the same z but opposite values of x^n − x^{−n} when n is even.",
            "original": "x = 1/2 and x = −2"
          },
          "slot2": {
            "description": "Field over which the polynomial coefficients are required; the argument works for any subfield of ℝ.",
            "original": "real coefficients"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}