{
  "index": "2001-A-3",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "For each integer $m$, consider the polynomial\n\\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\\] For what values of $m$ is $P_m(x)$\nthe product of two non-constant polynomials with integer coefficients?",
  "solution": "By the quadratic formula, if $P_m(x)=0$, then $x^2=m\\pm\n2\\sqrt{2m}+2$, and hence the four roots of $P_m$ are given by\n$S = \\{\\pm\\sqrt{m}\\pm\\sqrt{2}\\}$. If $P_m$ factors into two nonconstant\npolynomials over the integers, then some subset of $S$ consisting of one\nor two elements form the roots of a polynomial with integer coefficients.\n\nFirst suppose this subset has a single element, say $\\sqrt{m} \\pm \\sqrt{2}$;\nthis element must be a rational number.\nThen $(\\sqrt{m} \\pm \\sqrt{2})^2 = 2 + m \\pm  2 \\sqrt{2m}$ is an integer,\nso $m$ is twice a perfect square, say $m = 2n^2$. But then\n$\\sqrt{m} \\pm \\sqrt{2} = (n\\pm 1)\\sqrt{2}$ is only rational if $n=\\pm 1$,\ni.e., if $m = 2$.\n\nNext, suppose that the subset contains two elements; then we can take\nit to be one of $\\{\\sqrt{m} \\pm \\sqrt{2}\\}$, $\\{\\sqrt{2} \\pm \\sqrt{m}\\}$\nor $\\{\\pm (\\sqrt{m} + \\sqrt{2})\\}$. In all cases, the sum and the product\nof the elements of the\nsubset must be a rational number. In the first case, this means\n$2\\sqrt{m} \\in \\QQ$, so $m$ is a perfect square. In the second case,\nwe have $2 \\sqrt{2} \\in \\QQ$, contradiction. In the third case, we have\n$(\\sqrt{m} + \\sqrt{2})^2 \\in \\QQ$, or $m + 2 + 2\\sqrt{2m} \\in \\QQ$, which\nmeans that $m$ is twice a perfect square.\n\nWe conclude that $P_m(x)$ factors into two nonconstant polynomials over\nthe integers if and only if $m$ is either a square or twice a square.\n\nNote: a more sophisticated interpretation of this argument can be given\nusing Galois theory. Namely, if $m$ is neither a square nor twice a square,\nthen the number fields $\\QQ(\\sqrt{m})$ and $\\QQ(\\sqrt{2})$ are distinct\nquadratic fields, so their compositum is a number field of degree 4, whose\nGalois group acts transitively on $\\{\\pm \\sqrt{m} \\pm \\sqrt{2}\\}$. Thus\n$P_m$ is irreducible.",
  "vars": [
    "x",
    "n",
    "S"
  ],
  "params": [
    "m",
    "P_m"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variable",
        "n": "indexer",
        "S": "rootset",
        "m": "intparam",
        "P_m": "givenpolynomial"
      },
      "question": "For each integer $intparam$, consider the polynomial\n\\[givenpolynomial(variable)=variable^{4}-(2 intparam+4) variable^{2}+(intparam-2)^{2}.\\]\nFor what values of $intparam$ is $givenpolynomial(variable)$ the product of two non-constant polynomials with integer coefficients?",
      "solution": "By the quadratic formula, if $givenpolynomial(variable)=0$, then $variable^{2}=intparam\\pm2\\sqrt{2 intparam}+2$, and hence the four roots of $givenpolynomial$ are given by $rootset = \\{\\pm\\sqrt{intparam}\\pm\\sqrt{2}\\}$. If $givenpolynomial$ factors into two nonconstant polynomials over the integers, then some subset of $rootset$ consisting of one or two elements form the roots of a polynomial with integer coefficients.\n\nFirst suppose this subset has a single element, say $\\sqrt{intparam} \\pm \\sqrt{2}$; this element must be a rational number. Then $(\\sqrt{intparam} \\pm \\sqrt{2})^{2} = 2 + intparam \\pm 2 \\sqrt{2 intparam}$ is an integer, so $intparam$ is twice a perfect square, say $intparam = 2 indexer^{2}$. But then $\\sqrt{intparam} \\pm \\sqrt{2} = (indexer\\pm 1)\\sqrt{2}$ is only rational if $indexer=\\pm 1$, i.e., if $intparam = 2$.\n\nNext, suppose that the subset contains two elements; then we can take it to be one of $\\{\\sqrt{intparam} \\pm \\sqrt{2}\\}$, $\\{\\sqrt{2} \\pm \\sqrt{intparam}\\}$ or $\\{\\pm (\\sqrt{intparam} + \\sqrt{2})\\}$. In all cases, the sum and the product of the elements of the subset must be a rational number. In the first case, this means $2\\sqrt{intparam} \\in \\QQ$, so $intparam$ is a perfect square. In the second case, we have $2 \\sqrt{2} \\in \\QQ$, contradiction. In the third case, we have $(\\sqrt{intparam} + \\sqrt{2})^{2} \\in \\QQ$, or $intparam + 2 + 2\\sqrt{2 intparam} \\in \\QQ$, which means that $intparam$ is twice a perfect square.\n\nWe conclude that $givenpolynomial(variable)$ factors into two nonconstant polynomials over the integers if and only if $intparam$ is either a square or twice a square.\n\nNote: a more sophisticated interpretation of this argument can be given using Galois theory. Namely, if $intparam$ is neither a square nor twice a square, then the number fields $\\QQ(\\sqrt{intparam})$ and $\\QQ(\\sqrt{2})$ are distinct quadratic fields, so their compositum is a number field of degree 4, whose Galois group acts transitively on $\\{\\pm \\sqrt{intparam} \\pm \\sqrt{2}\\}$. Thus $givenpolynomial$ is irreducible."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "wandering",
        "n": "labyrinth",
        "S": "constell",
        "m": "parchment",
        "P_m": "riverdelta"
      },
      "question": "For each integer $parchment$, consider the polynomial\n\\[riverdelta(wandering)=wandering^4-(2parchment+4)wandering^2+(parchment-2)^2.\\] For what values of $parchment$ is $riverdelta(wandering)$\nthe product of two non-constant polynomials with integer coefficients?",
      "solution": "By the quadratic formula, if $riverdelta(wandering)=0$, then $wandering^2=parchment\\pm\n2\\sqrt{2parchment}+2$, and hence the four roots of $riverdelta$ are given by\n$constell = \\{\\pm\\sqrt{parchment}\\pm\\sqrt{2}\\}$. If $riverdelta$ factors into two nonconstant\npolynomials over the integers, then some subset of $constell$ consisting of one\nor two elements form the roots of a polynomial with integer coefficients.\n\nFirst suppose this subset has a single element, say $\\sqrt{parchment} \\pm \\sqrt{2}$;\nthis element must be a rational number.\nThen $(\\sqrt{parchment} \\pm \\sqrt{2})^2 = 2 + parchment \\pm  2 \\sqrt{2parchment}$ is an integer,\nso $parchment$ is twice a perfect square, say $parchment = 2labyrinth^2$. But then\n$\\sqrt{parchment} \\pm \\sqrt{2} = (labyrinth\\pm 1)\\sqrt{2}$ is only rational if $labyrinth=\\pm 1$,\ni.e., if $parchment = 2$.\n\nNext, suppose that the subset contains two elements; then we can take\nit to be one of $\\{\\sqrt{parchment} \\pm \\sqrt{2}\\}$, $\\{\\sqrt{2} \\pm \\sqrt{parchment}\\}$\nor $\\{\\pm (\\sqrt{parchment} + \\sqrt{2})\\}$. In all cases, the sum and the product\nof the elements of the\nsubset must be a rational number. In the first case, this means\n$2\\sqrt{parchment} \\in \\QQ$, so $parchment$ is a perfect square. In the second case,\nwe have $2 \\sqrt{2} \\in \\QQ$, contradiction. In the third case, we have\n$(\\sqrt{parchment} + \\sqrt{2})^2 \\in \\QQ$, or $parchment + 2 + 2\\sqrt{2parchment} \\in \\QQ$, which\nmeans that $parchment$ is twice a perfect square.\n\nWe conclude that $riverdelta(wandering)$ factors into two nonconstant polynomials over\nthe integers if and only if $parchment$ is either a square or twice a square.\n\nNote: a more sophisticated interpretation of this argument can be given\nusing Galois theory. Namely, if $parchment$ is neither a square nor twice a square,\nthen the number fields $\\QQ(\\sqrt{parchment})$ and $\\QQ(\\sqrt{2})$ are distinct\nquadratic fields, so their compositum is a number field of degree 4, whose\nGalois group acts transitively on $\\{\\pm \\sqrt{parchment} \\pm \\sqrt{2}\\}$. Thus\n$riverdelta$ is irreducible."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantvalue",
        "n": "boundless",
        "S": "singleton",
        "m": "fractional",
        "P_m": "antipolynomial"
      },
      "question": "For each integer $fractional$, consider the polynomial\n\\[antipolynomial(constantvalue)=constantvalue^4-(2fractional+4)constantvalue^2+(fractional-2)^2.\\] For what values of $fractional$ is $antipolynomial(constantvalue)$\nthe product of two non-constant polynomials with integer coefficients?",
      "solution": "By the quadratic formula, if $antipolynomial(constantvalue)=0$, then $constantvalue^2=fractional\\pm\n2\\sqrt{2fractional}+2$, and hence the four roots of $antipolynomial$ are given by\n$singleton = \\{\\pm\\sqrt{fractional}\\pm\\sqrt{2}\\}$. If $antipolynomial$ factors into two nonconstant\npolynomials over the integers, then some subset of $singleton$ consisting of one\nor two elements form the roots of a polynomial with integer coefficients.\n\nFirst suppose this subset has a single element, say $\\sqrt{fractional} \\pm \\sqrt{2}$;\nthis element must be a rational number.\nThen $(\\sqrt{fractional} \\pm \\sqrt{2})^2 = 2 + fractional \\pm  2 \\sqrt{2fractional}$ is an integer,\nso $fractional$ is twice a perfect square, say $fractional = 2boundless^2$. But then\n$\\sqrt{fractional} \\pm \\sqrt{2} = (boundless\\pm 1)\\sqrt{2}$ is only rational if $boundless=\\pm 1$,\ni.e., if $fractional = 2$.\n\nNext, suppose that the subset contains two elements; then we can take\nit to be one of $\\{\\sqrt{fractional} \\pm \\sqrt{2}\\}$, $\\{\\sqrt{2} \\pm \\sqrt{fractional}\\}$\nor $\\{\\pm (\\sqrt{fractional} + \\sqrt{2})\\}$. In all cases, the sum and the product\nof the elements of the\nsubset must be a rational number. In the first case, this means\n$2\\sqrt{fractional} \\in \\QQ$, so $fractional$ is a perfect square. In the second case,\nwe have $2 \\sqrt{2} \\in \\QQ$, contradiction. In the third case, we have\n$(\\sqrt{fractional} + \\sqrt{2})^2 \\in \\QQ$, or $fractional + 2 + 2\\sqrt{2fractional} \\in \\QQ$, which\nmeans that $fractional$ is twice a perfect square.\n\nWe conclude that $antipolynomial(constantvalue)$ factors into two nonconstant polynomials over\nthe integers if and only if $fractional$ is either a square or twice a square.\n\nNote: a more sophisticated interpretation of this argument can be given\nusing Galois theory. Namely, if $fractional$ is neither a square nor twice a square,\nthen the number fields $\\QQ(\\sqrt{fractional})$ and $\\QQ(\\sqrt{2})$ are distinct\nquadratic fields, so their compositum is a number field of degree 4, whose\nGalois group acts transitively on $\\{\\pm \\sqrt{fractional} \\pm \\sqrt{2}\\}$. Thus\n$antipolynomial$ is irreducible."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "n": "hjgrksla",
        "S": "vlmprgth",
        "m": "brzcpkdu",
        "P_m": "fjqhtlda"
      },
      "question": "For each integer $brzcpkdu$, consider the polynomial\n\\[fjqhtlda(qzxwvtnp)=qzxwvtnp^4-(2brzcpkdu+4)qzxwvtnp^2+(brzcpkdu-2)^2.\\] For what values of $brzcpkdu$ is $fjqhtlda(qzxwvtnp)$\nthe product of two non-constant polynomials with integer coefficients?",
      "solution": "By the quadratic formula, if $fjqhtlda(qzxwvtnp)=0$, then $qzxwvtnp^2=brzcpkdu\\pm\n2\\sqrt{2brzcpkdu}+2$, and hence the four roots of $fjqhtlda$ are given by\n$vlmprgth = \\{\\pm\\sqrt{brzcpkdu}\\pm\\sqrt{2}\\}$. If $fjqhtlda$ factors into two nonconstant\npolynomials over the integers, then some subset of $vlmprgth$ consisting of one\nor two elements form the roots of a polynomial with integer coefficients.\n\nFirst suppose this subset has a single element, say $\\sqrt{brzcpkdu} \\pm \\sqrt{2}$;\nthis element must be a rational number.\nThen $(\\sqrt{brzcpkdu} \\pm \\sqrt{2})^2 = 2 + brzcpkdu \\pm  2 \\sqrt{2brzcpkdu}$ is an integer,\nso $brzcpkdu$ is twice a perfect square, say $brzcpkdu = 2hjgrksla^2$. But then\n$\\sqrt{brzcpkdu} \\pm \\sqrt{2} = (hjgrksla\\pm 1)\\sqrt{2}$ is only rational if $hjgrksla=\\pm 1$,\ni.e., if $brzcpkdu = 2$.\n\nNext, suppose that the subset contains two elements; then we can take\nit to be one of $\\{\\sqrt{brzcpkdu} \\pm \\sqrt{2}\\}$, $\\{\\sqrt{2} \\pm \\sqrt{brzcpkdu}\\}$\nor $\\{\\pm (\\sqrt{brzcpkdu} + \\sqrt{2})\\}$. In all cases, the sum and the product\nof the elements of the\nsubset must be a rational number. In the first case, this means\n$2\\sqrt{brzcpkdu} \\in \\QQ$, so $brzcpkdu$ is a perfect square. In the second case,\nwe have $2 \\sqrt{2} \\in \\QQ$, contradiction. In the third case, we have\n$(\\sqrt{brzcpkdu} + \\sqrt{2})^2 \\in \\QQ$, or $brzcpkdu + 2 + 2\\sqrt{2brzcpkdu} \\in \\QQ$, which\nmeans that $brzcpkdu$ is twice a perfect square.\n\nWe conclude that $fjqhtlda(qzxwvtnp)$ factors into two nonconstant polynomials over\nthe integers if and only if $brzcpkdu$ is either a square or twice a square.\n\nNote: a more sophisticated interpretation of this argument can be given\nusing Galois theory. Namely, if $brzcpkdu$ is neither a square nor twice a square,\nthen the number fields $\\QQ(\\sqrt{brzcpkdu})$ and $\\QQ(\\sqrt{2})$ are distinct\nquadratic fields, so their compositum is a number field of degree 4, whose\nGalois group acts transitively on $\\{\\pm \\sqrt{brzcpkdu} \\pm \\sqrt{2}\\}$. Thus\n$fjqhtlda$ is irreducible."
    },
    "kernel_variant": {
      "question": "Let  \n\\[\nR_{m,n}(x)=x^{4}-2\\,(m+n+4)\\,x^{2}+(m-n)^{2}\\qquad(m,n\\in\\mathbb Z)\n\\]  \nbe a monic quartic depending on two independent integer parameters.  \nDetermine all ordered pairs $\\,(m,n)\\in\\mathbb Z^{2}\\,$ for which the polynomial $R_{m,n}(x)$ can be written as a product of two non–constant polynomials with coefficients in $\\mathbb Z$.\n\n────────────────────────────────────────",
      "solution": "Throughout we abbreviate  \n\\[\nR_{m,n}(x)=x^{4}-2(m+n+4)x^{2}+(m-n)^{2},\\qquad(m,n)\\in\\mathbb Z^{2}.\n\\]\n\nStep 1.  Reduction to the product of two even quadratics.  \nAssume that\n\\[\nR_{m,n}(x)=A(x)\\,B(x),\\qquad A,B\\in\\mathbb Z[x],\\;1\\le\\deg A,\\deg B\\le 3.\n\\tag{1}\n\\]\n\nBecause $R_{m,n}$ is even, the multiset of its irreducible factors is closed under $x\\mapsto -x$.  \nIf $g(x)\\in\\mathbb Z[x]$ is an irreducible divisor, then $g(-x)$ is also irreducible.\n\n•  If $\\deg g=1$ then $g(x)=x-t$ with $t\\in\\mathbb Z$ by the rational–root test; its mate is $g(-x)=-(x+t)$, so together they contribute the even quadratic $x^{2}-t^{2}$.\n\n•  If $\\deg g$ is odd and at least $3$, then $\\deg\\bigl(g(x)g(-x)\\bigr)\\ge6>4$, impossible.\n\nHence all linear factors come in pairs, and no irreducible factor of odd degree $\\ge3$ can occur.  Thus every factorisation of $R_{m,n}$ can be regrouped into the product of two even quadratics.  Consequently we may – and shall – write  \n\\[\nR_{m,n}(x)=(x^{2}+p x+q)\\,(x^{2}+r x+s),\\qquad p,q,r,s\\in\\mathbb Z,\n\\tag{2}\n\\]\nwith each quadratic non-constant.  (If $m=n$ one of the quadratics will be $x^{2}$; this still fits into (2) by taking $q=0$ or $s=0$.)\n\nStep 2.  Coefficient comparison.  \nExpanding (2) and matching with $R_{m,n}$ yields  \n\\begin{align}\np+r &=0, \\tag{3a}\\\\\nps+qr &=0, \\tag{3b}\\\\\nq+s+pr &=-2(m+n+4), \\tag{3c}\\\\\nqs &=(m-n)^{2}. \\tag{3d}\n\\end{align}\nEquation (3a) gives $r=-p$.  Inserting this into (3b) provides  \n\\[\np(s-q)=0. \\tag{4}\n\\]\nTwo disjoint cases arise.\n\n────────────────────────────────────────  \nCase A.  $p=0$ (thus $r=0$).\n\nThen the factorisation reads $(x^{2}+q)(x^{2}+s)$, and (3c)–(3d) boil down to  \n\\[\nq+s=-2(m+n+4),\\qquad q\\,s=(m-n)^{2}. \\tag{5}\n\\]\nEliminating $s$ gives the quadratic equation  \n\\[\nq^{2}+2(m+n+4)\\,q+(m-n)^{2}=0. \\tag{6}\n\\]\nIts discriminant equals  \n\\[\n\\Delta_q=4\\bigl((m+n+4)^{2}-(m-n)^{2}\\bigr)=16\\,(m+2)(n+2).\n\\]\nEquation (6) has an integer root precisely when  \n\\[\n(m+2)(n+2)\\text{ is a perfect square}. \\tag{A}\n\\]\nWriting $(m+2)(n+2)=k^{2}$ ($k\\ge0$) one obtains  \n\\[\nq=-\\bigl(m+n+4\\bigr)\\pm2k,\\qquad\ns=-\\bigl(m+n+4\\bigr)\\mp2k,\n\\]\nand hence  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}-m-n-4-2k\\bigr)\\,\n           \\bigl(x^{2}-m-n-4+2k\\bigr).\n\\tag{7}\n\\]\n\n────────────────────────────────────────  \nCase B.  $p\\neq0$.  Then (4) forces $s=q$ and $r=-p$.  Plugging this into (3c)–(3d) gives  \n\\begin{align}\np^{2}-2q &=2(m+n+4), \\tag{8a}\\\\\nq^{2} &=(m-n)^{2}. \\tag{8b}\n\\end{align}\nEquation (8b) yields $q=\\pm(m-n)$.\n\nB1.  $q=m-n$.  Equation (8a) becomes $p^{2}=4(m+2)$, so  \n\\[\nm+2\\text{ is a perfect square}. \\tag{B$_1$}\n\\]\nWrite $m+2=u^{2}$ with $u\\ge0$ and choose $p=\\pm2u$.  Then  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}+2u x+m-n\\bigr)\\,\n           \\bigl(x^{2}-2u x+m-n\\bigr).\n\\tag{9}\n\\]\n(The derivation assumed $p\\neq0$, hence $u\\neq0$.  If $u=0$ (i.e.\\ $m=-2$) then we are really in Case A, see below.)\n\nB2.  $q=n-m$.  Equation (8a) gives $p^{2}=4(n+2)$, so  \n\\[\nn+2\\text{ is a perfect square}. \\tag{B$_2$}\n\\]\nWrite $n+2=v^{2}$ with $v\\ge0$ and select $p=\\pm2v$.  Then  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}+2v x+n-m\\bigr)\\,\n           \\bigl(x^{2}-2v x+n-m\\bigr).\n\\tag{10}\n\\]\n(Again, if $v=0$ (that is, $n=-2$) we are in Case A.)\n\n────────────────────────────────────────  \nStep 3.  Consolidation of the conditions.  \n\nNecessity.  \nEvery integral factorisation of $R_{m,n}$ belongs to Case A or B, and therefore forces at least one of the conditions  \n(A), (B$_1$), (B$_2$).\n\nSufficiency.  \nConversely, whenever (A), (B$_1$) or (B$_2$) holds, the explicit decompositions (7), (9) or (10) furnish the desired factorisation into two non-constant polynomials in $\\mathbb Z[x]$.\n\nThus  \n\\[\n\\boxed{\\;\nR_{m,n}\\text{ is reducible in }\\mathbb Z[x]\\;\n\\Longleftrightarrow\\;\n(m+2)(n+2)\\text{ is a perfect square}\\;\n\\text{or}\\;\nm+2\\text{ is a perfect square}\\;\n\\text{or}\\;\nn+2\\text{ is a perfect square}.}\n\\]\n\n────────────────────────────────────────  \nStep 4.  Reformulation via square–free parts.  \n\nIf $m\\neq-2$ write $m+2=d\\,r^{2}$ with $d$ square-free, $r\\ge0$; if $m=-2$ set $d=0$, $r=0$.  \nAnalogously, write $n+2=e\\,s^{2}$ with $e$ square-free ($e=0$ if $n=-2$).  Then  \n\\[\n\\begin{aligned}\n\\text{(A)}&\\Longleftrightarrow d=e,\\\\\n\\text{(B$_1$)}&\\Longleftrightarrow d=1,\\\\\n\\text{(B$_2$)}&\\Longleftrightarrow e=1.\n\\end{aligned}\n\\]\nHence $R_{m,n}$ is reducible over $\\mathbb Z$ precisely when the square-free parts of $m+2$ and $n+2$ coincide (possibly both $0$) or at least one of them equals $1$.\n\n────────────────────────────────────────  \nStep 5.  The special case $m=n$.  \n\nWhen $m=n$, the constant term of $R_{m,n}$ vanishes, so $x=0$ is a double root and  \n\\[\nR_{m,m}(x)=x^{2}\\bigl(x^{2}-4(m+2)\\bigr).\n\\]\nHere $p=0$; hence this decomposition falls under Case A (indeed condition (A) holds automatically because $(m+2)(n+2)=(m+2)^{2}$ is a perfect square).  If $m+2$ is itself a perfect square, then the second quadratic further factors according to Case B$_1$; nevertheless the global criterion above already covers every possibility.\n\n\\hfill$\\square$\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.772387",
        "was_fixed": false,
        "difficulty_analysis": "•  Higher dimensional parameter space – the problem now involves two independent integers instead of one, turning the classification from a single list into the study of a two-dimensional lattice of solutions.  \n•  More intricate algebraic structure – the field generated by the roots is a compositum of two quadratic extensions $\\Bbb Q(\\sqrt{m+2})$ and $\\Bbb Q(\\sqrt{n+2})$.  Determining reducibility forces the solver to analyse when this compositum has degree $2$ (or less) rather than $4$, an explicit use of Galois-theoretic ideas far deeper than in the original statement.  \n•  Multiple interacting cases – three genuinely different pairings of the eight conjugates must be examined and consolidated, each producing its own arithmetic condition on $(m,n)$.  \n•  Non-trivial number-theory – deciding when expressions like $(m+2)(n+2)$ are squares and translating these conditions into square-free language demands familiarity with classical results on quadratic fields and square-free factorisation.  \n•  Expanded solution length – compared with the original single-parameter quartic, the enhanced variant requires a full classification in $\\Bbb Z^{2}$, careful case-by-case root pairing, and explicit factorisations in each scenario, all of which substantially lengthen and complicate the argument."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \n\\[\nR_{m,n}(x)=x^{4}-2\\,(m+n+4)\\,x^{2}+(m-n)^{2}\\qquad(m,n\\in\\mathbb Z)\n\\]  \nbe a monic quartic depending on two independent integer parameters.  \nDetermine all ordered pairs $\\,(m,n)\\in\\mathbb Z^{2}\\,$ for which the polynomial $R_{m,n}(x)$ can be written as a product of two non–constant polynomials with coefficients in $\\mathbb Z$.\n\n────────────────────────────────────────",
      "solution": "Throughout we abbreviate  \n\\[\nR_{m,n}(x)=x^{4}-2(m+n+4)x^{2}+(m-n)^{2},\\qquad(m,n)\\in\\mathbb Z^{2}.\n\\]\n\nStep 1.  Reduction to the product of two even quadratics.  \nAssume that\n\\[\nR_{m,n}(x)=A(x)\\,B(x),\\qquad A,B\\in\\mathbb Z[x],\\;1\\le\\deg A,\\deg B\\le 3.\n\\tag{1}\n\\]\n\nBecause $R_{m,n}$ is even, the multiset of its irreducible factors is closed under $x\\mapsto -x$.  \nIf $g(x)\\in\\mathbb Z[x]$ is an irreducible divisor, then $g(-x)$ is also irreducible.\n\n•  If $\\deg g=1$ then $g(x)=x-t$ with $t\\in\\mathbb Z$ by the rational–root test; its mate is $g(-x)=-(x+t)$, so together they contribute the even quadratic $x^{2}-t^{2}$.\n\n•  If $\\deg g$ is odd and at least $3$, then $\\deg\\bigl(g(x)g(-x)\\bigr)\\ge6>4$, impossible.\n\nHence all linear factors come in pairs, and no irreducible factor of odd degree $\\ge3$ can occur.  Thus every factorisation of $R_{m,n}$ can be regrouped into the product of two even quadratics.  Consequently we may – and shall – write  \n\\[\nR_{m,n}(x)=(x^{2}+p x+q)\\,(x^{2}+r x+s),\\qquad p,q,r,s\\in\\mathbb Z,\n\\tag{2}\n\\]\nwith each quadratic non-constant.  (If $m=n$ one of the quadratics will be $x^{2}$; this still fits into (2) by taking $q=0$ or $s=0$.)\n\nStep 2.  Coefficient comparison.  \nExpanding (2) and matching with $R_{m,n}$ yields  \n\\begin{align}\np+r &=0, \\tag{3a}\\\\\nps+qr &=0, \\tag{3b}\\\\\nq+s+pr &=-2(m+n+4), \\tag{3c}\\\\\nqs &=(m-n)^{2}. \\tag{3d}\n\\end{align}\nEquation (3a) gives $r=-p$.  Inserting this into (3b) provides  \n\\[\np(s-q)=0. \\tag{4}\n\\]\nTwo disjoint cases arise.\n\n────────────────────────────────────────  \nCase A.  $p=0$ (thus $r=0$).\n\nThen the factorisation reads $(x^{2}+q)(x^{2}+s)$, and (3c)–(3d) boil down to  \n\\[\nq+s=-2(m+n+4),\\qquad q\\,s=(m-n)^{2}. \\tag{5}\n\\]\nEliminating $s$ gives the quadratic equation  \n\\[\nq^{2}+2(m+n+4)\\,q+(m-n)^{2}=0. \\tag{6}\n\\]\nIts discriminant equals  \n\\[\n\\Delta_q=4\\bigl((m+n+4)^{2}-(m-n)^{2}\\bigr)=16\\,(m+2)(n+2).\n\\]\nEquation (6) has an integer root precisely when  \n\\[\n(m+2)(n+2)\\text{ is a perfect square}. \\tag{A}\n\\]\nWriting $(m+2)(n+2)=k^{2}$ ($k\\ge0$) one obtains  \n\\[\nq=-\\bigl(m+n+4\\bigr)\\pm2k,\\qquad\ns=-\\bigl(m+n+4\\bigr)\\mp2k,\n\\]\nand hence  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}-m-n-4-2k\\bigr)\\,\n           \\bigl(x^{2}-m-n-4+2k\\bigr).\n\\tag{7}\n\\]\n\n────────────────────────────────────────  \nCase B.  $p\\neq0$.  Then (4) forces $s=q$ and $r=-p$.  Plugging this into (3c)–(3d) gives  \n\\begin{align}\np^{2}-2q &=2(m+n+4), \\tag{8a}\\\\\nq^{2} &=(m-n)^{2}. \\tag{8b}\n\\end{align}\nEquation (8b) yields $q=\\pm(m-n)$.\n\nB1.  $q=m-n$.  Equation (8a) becomes $p^{2}=4(m+2)$, so  \n\\[\nm+2\\text{ is a perfect square}. \\tag{B$_1$}\n\\]\nWrite $m+2=u^{2}$ with $u\\ge0$ and choose $p=\\pm2u$.  Then  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}+2u x+m-n\\bigr)\\,\n           \\bigl(x^{2}-2u x+m-n\\bigr).\n\\tag{9}\n\\]\n(The derivation assumed $p\\neq0$, hence $u\\neq0$.  If $u=0$ (i.e.\\ $m=-2$) then we are really in Case A, see below.)\n\nB2.  $q=n-m$.  Equation (8a) gives $p^{2}=4(n+2)$, so  \n\\[\nn+2\\text{ is a perfect square}. \\tag{B$_2$}\n\\]\nWrite $n+2=v^{2}$ with $v\\ge0$ and select $p=\\pm2v$.  Then  \n\\[\nR_{m,n}(x)=\\bigl(x^{2}+2v x+n-m\\bigr)\\,\n           \\bigl(x^{2}-2v x+n-m\\bigr).\n\\tag{10}\n\\]\n(Again, if $v=0$ (that is, $n=-2$) we are in Case A.)\n\n────────────────────────────────────────  \nStep 3.  Consolidation of the conditions.  \n\nNecessity.  \nEvery integral factorisation of $R_{m,n}$ belongs to Case A or B, and therefore forces at least one of the conditions  \n(A), (B$_1$), (B$_2$).\n\nSufficiency.  \nConversely, whenever (A), (B$_1$) or (B$_2$) holds, the explicit decompositions (7), (9) or (10) furnish the desired factorisation into two non-constant polynomials in $\\mathbb Z[x]$.\n\nThus  \n\\[\n\\boxed{\\;\nR_{m,n}\\text{ is reducible in }\\mathbb Z[x]\\;\n\\Longleftrightarrow\\;\n(m+2)(n+2)\\text{ is a perfect square}\\;\n\\text{or}\\;\nm+2\\text{ is a perfect square}\\;\n\\text{or}\\;\nn+2\\text{ is a perfect square}.}\n\\]\n\n────────────────────────────────────────  \nStep 4.  Reformulation via square–free parts.  \n\nIf $m\\neq-2$ write $m+2=d\\,r^{2}$ with $d$ square-free, $r\\ge0$; if $m=-2$ set $d=0$, $r=0$.  \nAnalogously, write $n+2=e\\,s^{2}$ with $e$ square-free ($e=0$ if $n=-2$).  Then  \n\\[\n\\begin{aligned}\n\\text{(A)}&\\Longleftrightarrow d=e,\\\\\n\\text{(B$_1$)}&\\Longleftrightarrow d=1,\\\\\n\\text{(B$_2$)}&\\Longleftrightarrow e=1.\n\\end{aligned}\n\\]\nHence $R_{m,n}$ is reducible over $\\mathbb Z$ precisely when the square-free parts of $m+2$ and $n+2$ coincide (possibly both $0$) or at least one of them equals $1$.\n\n────────────────────────────────────────  \nStep 5.  The special case $m=n$.  \n\nWhen $m=n$, the constant term of $R_{m,n}$ vanishes, so $x=0$ is a double root and  \n\\[\nR_{m,m}(x)=x^{2}\\bigl(x^{2}-4(m+2)\\bigr).\n\\]\nHere $p=0$; hence this decomposition falls under Case A (indeed condition (A) holds automatically because $(m+2)(n+2)=(m+2)^{2}$ is a perfect square).  If $m+2$ is itself a perfect square, then the second quadratic further factors according to Case B$_1$; nevertheless the global criterion above already covers every possibility.\n\n\\hfill$\\square$\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.591544",
        "was_fixed": false,
        "difficulty_analysis": "•  Higher dimensional parameter space – the problem now involves two independent integers instead of one, turning the classification from a single list into the study of a two-dimensional lattice of solutions.  \n•  More intricate algebraic structure – the field generated by the roots is a compositum of two quadratic extensions $\\Bbb Q(\\sqrt{m+2})$ and $\\Bbb Q(\\sqrt{n+2})$.  Determining reducibility forces the solver to analyse when this compositum has degree $2$ (or less) rather than $4$, an explicit use of Galois-theoretic ideas far deeper than in the original statement.  \n•  Multiple interacting cases – three genuinely different pairings of the eight conjugates must be examined and consolidated, each producing its own arithmetic condition on $(m,n)$.  \n•  Non-trivial number-theory – deciding when expressions like $(m+2)(n+2)$ are squares and translating these conditions into square-free language demands familiarity with classical results on quadratic fields and square-free factorisation.  \n•  Expanded solution length – compared with the original single-parameter quartic, the enhanced variant requires a full classification in $\\Bbb Z^{2}$, careful case-by-case root pairing, and explicit factorisations in each scenario, all of which substantially lengthen and complicate the argument."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}