{
  "index": "1988-B-1",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "A \\emph{composite} (positive integer) is a product $ab$ with $a$ and\n$b$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that\nevery composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive\nintegers.",
  "solution": "Solution. Substituting \\( z=1 \\) yields \\( (x+1)(y+1) \\), so to represent the composite number \\( n=a b \\) with \\( a, b \\geq 2 \\), let \\( (x, y, z)=(a-1, b-1,1) \\).\n\nRemark. Although the problem asks only about representing composite numbers, all but finitely many prime numbers are representable too. Theorem 1.1 of [BC] proves that the only positive integers not of the form \\( x y+x z+y z+1 \\) for integers \\( x, y, z>0 \\) are the 19 integers \\( 1,2,3,5,7,11,19,23,31,43,59,71,79,103,131,191,211,331 \\), and 463 , and possibly a 20 th integer greater than \\( 10^{11} \\). Moreover, if the Generalized Riemann Hypothesis (GRH) is true, then the 20th integer does not exist. (See [Le] for earlier work on this problem.)\n\nThe situation is analogous to that of the class number 1 problem: for many years it was known that the squarefree integers \\( d>0 \\) such that \\( \\mathbb{Q}(\\sqrt{-d}) \\) has class number 1 were\n\\[\nd=1,2,3,7,11,19,43,67,163\n\\]\nand possibly one more; the existence of this tenth imaginary quadratic field of class number 1 was eventually ruled out: see the appendix to [ Se 3\\( ] \\) for the history and the connection of this problem to integer points on modular curves.\n\nIn fact, researchers in the 19th century connected the problem of determining the positive integers representable by \\( x y+x z+y z+1 \\) to problems about class numbers of quadratic imaginary fields, or equivalently class numbers of binary quadratic forms: [Mord1] mentions that the connection is present in comments by Liouville, in Jour. de maths., series 2, tome 7, 1862, page 44, on a paper by Hermite. See also [Bel],\n[Wh], and [Mord2, p. 291]. The GRH implies the nonexistence of a Siegel zero for the Dirichlet \\( L \\)-functions associated to these fields, and this is what is used in the proof of Theorem 1.1 of [BC].",
  "vars": [
    "a",
    "b",
    "x",
    "y",
    "z",
    "n",
    "d"
  ],
  "params": [
    "L",
    "Q"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "a": "firstfactor",
        "b": "secondfactor",
        "x": "firstvar",
        "y": "secondvar",
        "z": "thirdvar",
        "n": "composite",
        "d": "squarefree",
        "L": "dirichlet",
        "Q": "rationals"
      },
      "question": "A \\emph{composite} (positive integer) is a product $firstfactor secondfactor$ with $firstfactor$ and $secondfactor$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that every composite is expressible as $firstvar secondvar + firstvar thirdvar + secondvar thirdvar + 1$, with $firstvar, secondvar, thirdvar$ positive integers.",
      "solution": "Solution. Substituting \\( thirdvar = 1 \\) yields \\( (firstvar+1)(secondvar+1) \\), so to represent the composite number \\( composite = firstfactor secondfactor \\) with \\( firstfactor, secondfactor \\ge 2 \\), let \\( (firstvar, secondvar, thirdvar) = (firstfactor-1, secondfactor-1, 1) \\).\n\nRemark. Although the problem asks only about representing composite numbers, all but finitely many prime numbers are representable too. Theorem 1.1 of [BC] proves that the only positive integers not of the form \\( firstvar secondvar + firstvar thirdvar + secondvar thirdvar + 1 \\) for integers \\( firstvar, secondvar, thirdvar > 0 \\) are the 19 integers \\( 1, 2, 3, 5, 7, 11, 19, 23, 31, 43, 59, 71, 79, 103, 131, 191, 211, 331 \\), and 463, and possibly a 20th integer greater than \\( 10^{11} \\).\n\nThe situation is analogous to that of the class number 1 problem: for many years it was known that the squarefree integers \\( squarefree > 0 \\) such that \\( \\mathbb{rationals}(\\sqrt{-squarefree}) \\) has class number 1 were\n\\[\n squarefree = 1, 2, 3, 7, 11, 19, 43, 67, 163\n\\]\nand possibly one more; the existence of this tenth imaginary quadratic field of class number 1 was eventually ruled out: see the appendix to [ Se 3\\( ] \\) for the history and the connection of this problem to integer points on modular curves.\n\nIn fact, researchers in the 19th century connected the problem of determining the positive integers representable by \\( firstvar secondvar + firstvar thirdvar + secondvar thirdvar + 1 \\) to problems about class numbers of quadratic imaginary fields, or equivalently class numbers of binary quadratic forms: [Mord1] mentions that the connection is present in comments by Liouville, in Jour. de maths., series 2, tome 7, 1862, page 44, on a paper by Hermite. See also [Bel], [Wh], and [Mord2, p. 291]. The GRH implies the nonexistence of a Siegel zero for the Dirichlet \\( dirichlet \\)-functions associated to these fields, and this is what is used in the proof of Theorem 1.1 of [BC]."
    },
    "descriptive_long_confusing": {
      "map": {
        "a": "raincloud",
        "b": "stargazer",
        "x": "moonstone",
        "y": "riverbank",
        "z": "tidalwave",
        "n": "huckleberry",
        "d": "broomstick",
        "L": "marshmallow",
        "Q": "watermelon"
      },
      "question": "A \\emph{composite} (positive integer) is a product $raincloud stargazer$ with $raincloud$ and\n$stargazer$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that\nevery composite is expressible as $moonstone riverbank + moonstone tidalwave + riverbank tidalwave + 1$, with $moonstone, riverbank, tidalwave$ positive\nintegers.",
      "solution": "Solution. Substituting \\( tidalwave = 1 \\) yields \\( (moonstone+1)(riverbank+1) \\), so to represent the composite number \\( huckleberry = raincloud stargazer \\) with \\( raincloud, stargazer \\geq 2 \\), let \\( (moonstone, riverbank, tidalwave)=(raincloud-1, stargazer-1,1) \\).\n\nRemark. Although the problem asks only about representing composite numbers, all but finitely many prime numbers are representable too. Theorem 1.1 of [BC] proves that the only positive integers not of the form \\( moonstone riverbank + moonstone tidalwave + riverbank tidalwave + 1 \\) for integers \\( moonstone, riverbank, tidalwave>0 \\) are the 19 integers \\( 1,2,3,5,7,11,19,23,31,43,59,71,79,103,131,191,211,331 \\), and 463 , and possibly a 20 th integer greater than \\( 10^{11} \\). Moreover, if the Generalized Riemann Hypothesis (GRH) is true, then the 20th integer does not exist. (See [Le] for earlier work on this problem.)\n\nThe situation is analogous to that of the class number 1 problem: for many years it was known that the squarefree integers \\( broomstick>0 \\) such that \\( \\mathbb{Q}(\\sqrt{-broomstick}) \\) has class number 1 were\n\\[\nbroomstick=1,2,3,7,11,19,43,67,163\n\\]\nand possibly one more; the existence of this tenth imaginary quadratic field of class number 1 was eventually ruled out: see the appendix to [ Se 3\\( ] \\) for the history and the connection of this problem to integer points on modular curves.\n\nIn fact, researchers in the 19th century connected the problem of determining the positive integers representable by \\( moonstone riverbank + moonstone tidalwave + riverbank tidalwave + 1 \\) to problems about class numbers of quadratic imaginary fields, or equivalently class numbers of binary quadratic forms: [Mord1] mentions that the connection is present in comments by Liouville, in Jour. de maths., series 2, tome 7, 1862, page 44, on a paper by Hermite. See also [Bel],\n[Wh], and [Mord2, p. 291]. The GRH implies the nonexistence of a Siegel zero for the Dirichlet \\( marshmallow \\)-functions associated to these fields, and this is what is used in the proof of Theorem 1.1 of [BC]."
    },
    "descriptive_long_misleading": {
      "map": {
        "a": "endingchar",
        "b": "startingpt",
        "x": "finalvalue",
        "y": "initialval",
        "z": "alphasymbol",
        "n": "primevalue",
        "d": "squarefull",
        "L": "nonseries",
        "Q": "irrational"
      },
      "question": "A \\emph{composite} (positive integer) is a product $endingchar startingpt$ with $endingchar$ and\n$startingpt$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that\nevery composite is expressible as $finalvalue initialval+finalvalue alphasymbol+initialval alphasymbol+1$, with $finalvalue,initialval,alphasymbol$ positive\nintegers.",
      "solution": "Solution. Substituting \\( alphasymbol=1 \\) yields \\( (finalvalue+1)(initialval+1) \\), so to represent the composite number \\( primevalue=endingchar startingpt \\) with \\( endingchar, startingpt \\geq 2 \\), let \\( (finalvalue, initialval, alphasymbol)=(endingchar-1, startingpt-1,1) \\).\n\nRemark. Although the problem asks only about representing composite numbers, all but finitely many prime numbers are representable too. Theorem 1.1 of [BC] proves that the only positive integers not of the form \\( finalvalue initialval+finalvalue alphasymbol+initialval alphasymbol+1 \\) for integers \\( finalvalue, initialval, alphasymbol>0 \\) are the 19 integers \\( 1,2,3,5,7,11,19,23,31,43,59,71,79,103,131,191,211,331 \\), and 463 , and possibly a 20 th integer greater than \\( 10^{11} \\). Moreover, if the Generalized Riemann Hypothesis (GRH) is true, then the 20th integer does not exist. (See [Le] for earlier work on this problem.)\n\nThe situation is analogous to that of the class number 1 problem: for many years it was known that the squarefree integers \\( squarefull>0 \\) such that \\( \\mathbb{irrational}(\\sqrt{-squarefull}) \\) has class number 1 were\n\\[\nsquarefull=1,2,3,7,11,19,43,67,163\n\\]\nand possibly one more; the existence of this tenth imaginary quadratic field of class number 1 was eventually ruled out: see the appendix to [ Se 3\\( ] \\) for the history and the connection of this problem to integer points on modular curves.\n\nIn fact, researchers in the 19th century connected the problem of determining the positive integers representable by \\( finalvalue initialval+finalvalue alphasymbol+initialval alphasymbol+1 \\) to problems about class numbers of quadratic imaginary fields, or equivalently class numbers of binary quadratic forms: [Mord1] mentions that the connection is present in comments by Liouville, in Jour. de maths., series 2, tome 7, 1862, page 44, on a paper by Hermite. See also [Bel],\n[Wh], and [Mord2, p. 291]. The GRH implies the nonexistence of a Siegel zero for the Dirichlet \\( nonseries \\)-functions associated to these fields, and this is what is used in the proof of Theorem 1.1 of [BC]."
    },
    "garbled_string": {
      "map": {
        "a": "qzxwvtnp",
        "b": "hjgrksla",
        "x": "pmcovfdy",
        "y": "rltgensw",
        "z": "kqspvham",
        "n": "wczhmyet",
        "d": "fnxqbria",
        "L": "ogtjwmea",
        "Q": "rmbzclou"
      },
      "question": "A \\emph{composite} (positive integer) is a product $qzxwvtnp hjgrksla$ with $qzxwvtnp$ and\n$hjgrksla$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that\nevery composite is expressible as $pmcovfdyrltgensw+pmcovfdykqspvham+rltgenswkqspvham+1$, with $pmcovfdy,rltgensw,kqspvham$ positive\nintegers.",
      "solution": "Solution. Substituting \\( kqspvham=1 \\) yields \\( (pmcovfdy+1)(rltgensw+1) \\), so to represent the composite number \\( wczhmyet=qzxwvtnp hjgrksla \\) with \\( qzxwvtnp, hjgrksla \\geq 2 \\), let \\( (pmcovfdy, rltgensw, kqspvham)=(qzxwvtnp-1, hjgrksla-1,1) \\).\n\nRemark. Although the problem asks only about representing composite numbers, all but finitely many prime numbers are representable too. Theorem 1.1 of [BC] proves that the only positive integers not of the form \\( pmcovfdyrltgensw+pmcovfdykqspvham+rltgenswkqspvham+1 \\) for integers \\( pmcovfdy, rltgensw, kqspvham>0 \\) are the 19 integers \\( 1,2,3,5,7,11,19,23,31,43,59,71,79,103,131,191,211,331 \\), and 463 , and possibly a 20 th integer greater than \\( 10^{11} \\). Moreover, if the Generalized Riemann Hypothesis (GRH) is true, then the 20th integer does not exist. (See [Le] for earlier work on this problem.)\n\nThe situation is analogous to that of the class number 1 problem: for many years it was known that the squarefree integers \\( fnxqbria>0 \\) such that \\( rmbzclou(\\sqrt{-fnxqbria}) \\) has class number 1 were\n\\[\nfnxqbria=1,2,3,7,11,19,43,67,163\n\\]\nand possibly one more; the existence of this tenth imaginary quadratic field of class number 1 was eventually ruled out: see the appendix to [ Se 3\\( ] \\) for the history and the connection of this problem to integer points on modular curves.\n\nIn fact, researchers in the 19th century connected the problem of determining the positive integers representable by \\( pmcovfdyrltgensw+pmcovfdykqspvham+rltgenswkqspvham+1 \\) to problems about class numbers of quadratic imaginary fields, or equivalently class numbers of binary quadratic forms: [Mord1] mentions that the connection is present in comments by Liouville, in Jour. de maths., series 2, tome 7, 1862, page 44, on a paper by Hermite. See also [Bel],\n[Wh], and [Mord2, p. 291]. The GRH implies the nonexistence of a Siegel zero for the Dirichlet \\( ogtjwmea \\)-functions associated to these fields, and this is what is used in the proof of Theorem 1.1 of [BC]."
    },
    "kernel_variant": {
      "question": "Let \\(n\\) be a composite positive integer.  Prove that there exist positive integers \\(y\\) and \\(z\\) such that\n\\[\n n \\,=\\, yz + y + z + 1.\n\\]",
      "solution": "Because n is composite, we can write it as a product\nn = a b,   a,b ≥ 2.\nFix a third variable x to be 1; then for any positive integers y,z we have the identity\nxy + xz + yz + 1 = 1⋅y + 1⋅z + yz + 1 = y + z + yz + 1 = (y+1)(z+1).\nChoose\ny = a−1,\nz = b−1,\nwhich are positive because a,b ≥ 2. Substituting these choices yields\nyz + y + z + 1 = (a−1)(b−1) + (a−1) + (b−1) + 1 = ab = n.\nThus the required representation of n is obtained, completing the proof.",
      "_meta": {
        "core_steps": [
          "Fix one variable to 1 (e.g. set z = 1).",
          "Note that the form collapses: xy + x + y + 1 = (x + 1)(y + 1).",
          "Express the composite n as a·b with a, b ≥ 2.",
          "Take x = a − 1 and y = b − 1 (with the fixed variable = 1).",
          "Conclude n = (x + 1)(y + 1) = xy + xz + yz + 1."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Which of the three variables is frozen to the value 1 in step 1.",
            "original": "z"
          },
          "slot2": {
            "description": "Names / ordering of the two free variables that become a−1 and b−1.",
            "original": "(x, y)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}