{
  "index": "1950-A-5",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "5. A function \\( D(n) \\) of the positive integral variable \\( n \\) is defined by the following properties: \\( D(1)=0, D(p)=1 \\) if \\( p \\) is a prime, \\( D(u v)=u D(v)+ \\) \\( v D(u) \\) for any two positive integers \\( u \\) and \\( v \\). Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely \\( \\boldsymbol{D}(n) \\). (Derive a formula for \\( \\boldsymbol{D}(n) / n \\), assuming that \\( n=p_{1}^{\\alpha_{1}} p_{2}{ }^{\\alpha_{2}} \\cdots p_{k}{ }^{\\alpha_{k}} \\) where \\( p_{1}, p_{2}, \\ldots, p_{k} \\) are different primes.)\n(ii) For what values of \\( n \\) is \\( D(n)=n \\) ?\n(iii) Define \\( D^{2}(n)=D[D(n)] \\), etc., and find the limit of \\( D^{m}(63) \\) as \\( m \\) tends to \\( \\infty \\).",
  "solution": "Solution. (i) Suppose there is a function \\( D \\) with the required properties. We have\n\\[\n\\frac{D(u v)}{u v}=\\frac{D(u)}{u}+\\frac{D(v)}{v}\n\\]\nand by induction\n\\[\n\\frac{D\\left(u_{1} u_{2} \\cdots u_{k}\\right)}{u_{1} u_{2} \\cdots u_{k}}=\\sum_{i} \\frac{D\\left(u_{i}\\right)}{u_{i}} .\n\\]\n\nHence\n\\[\n\\frac{D\\left(p^{\\alpha}\\right)}{p^{\\alpha}}=\\alpha \\frac{D(p)}{p}=\\frac{\\alpha}{p}\n\\]\nif \\( p \\) is a prime, and for any integer \\( n \\) with prime factorization \\( p_{1}{ }^{\\alpha_{1}} p_{2}{ }^{\\alpha_{2}} \\cdots p_{k}{ }^{\\alpha_{k}} \\) we have\n\\[\n\\frac{D(n)}{n}=\\sum_{i} \\frac{\\alpha_{i}}{p_{i}}\n\\]\n\nThis equation shows that there is at most one function with the given properties.\n\nOn the other hand, since every integer \\( n>1 \\) has a factorization into primes that is unique apart from order, and since the order in which the primes are numbered does not affect the sum in (2), we can define \\( D(1) \\) \\( =0 \\) and use (2) to define \\( D(n) \\) for \\( n>1 \\). So defined, \\( D(p)=1 \\) for \\( p \\) a prime, and \\( D \\) clearly satisfies (1), which is equivalent to \\( D(u v)=u D(v)+ \\) \\( v D(u) \\). Thus there is a unique function with the prescribed properties.\n\nWe note for future reference that \\( D(n)>0 \\) for \\( n>1 \\).\n(ii) The equation \\( D(n)=n \\) is equivalent to\n\\[\n\\frac{\\alpha_{1}}{p_{1}}+\\frac{\\alpha_{2}}{p_{2}}+\\cdots+\\frac{\\alpha_{k}}{p_{k}}=1,\n\\]\nwhere \\( n=p_{1}{ }^{\\alpha_{1}} p_{2}{ }^{\\alpha_{2}} \\cdots p_{k}{ }^{\\alpha k} \\) is the prime factorization of \\( n \\). If (3) is multiplied through by \\( p_{1} p_{2} \\cdots p_{k-1} \\), we see that \\( p_{1} p_{2} \\cdots p_{k-1} \\alpha_{k} / p_{k} \\) is an integer. Since the \\( p \\) 's are all different, we conclude that \\( p_{k} \\) divides \\( \\alpha_{k} \\). So \\( \\alpha_{k} / p_{k} \\) is an integer, and it is clear from (3) that \\( k=1 \\) and \\( \\alpha_{k}=p_{k} \\). Thus any solution of \\( D(n)=n \\) has the form \\( n=p^{p} \\) where \\( p \\) is prime. Conversely, any such \\( n \\) is a solution.\n\\[\n\\begin{aligned}\nD(63) & =51, \\\\\nD^{2}(63) & =D(51)=20, \\\\\nD^{3}(63) & =D(20)=24, \\\\\nD^{4}(63) & =D(24)=44, \\\\\nD^{5}(63) & =D(44)=48\n\\end{aligned}\n\\]\n\nIt appears that \\( D^{m}(63) \\) has started to increase with \\( m \\).\nSuppose \\( n=4 k \\) where \\( k>1 \\). Then \\( D(n)=D(4) k+4 D(k)=4(k+ \\) \\( D(k))>4 k=n \\). Thus, if \\( n>4 \\) and \\( n \\) is divisible by 4 , then \\( D(n)>n \\) and \\( D(n) \\) is divisible by 4 . This implies that the sequence\n\\[\nD(n), D^{2}(n), D^{3}(n), \\ldots, D^{m}(n), \\ldots\n\\]\nis strictly increasing. Since \\( D \\) takes integral values, \\( D^{m}(n) \\rightarrow \\infty \\) as \\( m \\rightarrow \\infty \\) whenever \\( n=4 k, k>1 \\).\n\nApplying this result to the case above, we see that \\( D^{m}(63)=D^{m-2}(20) \\)\n\\( \\rightarrow \\infty \\) as \\( m \\rightarrow \\infty \\).",
  "vars": [
    "n",
    "p",
    "k",
    "m",
    "u",
    "v",
    "u_1",
    "u_2",
    "u_k",
    "p_1",
    "p_2",
    "p_k",
    "\\\\alpha",
    "\\\\alpha_1",
    "\\\\alpha_2",
    "\\\\alpha_k"
  ],
  "params": [
    "D"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "intnum",
        "p": "primenum",
        "k": "indexer",
        "m": "iterate",
        "u": "firstvar",
        "v": "secondv",
        "u_1": "unitone",
        "u_2": "unittwo",
        "u_k": "unitvar",
        "p_1": "primeone",
        "p_2": "primetwo",
        "p_k": "primeidx",
        "\\alpha": "exponent",
        "\\alpha_1": "expoone",
        "\\alpha_2": "expotwo",
        "\\alpha_k": "expovar",
        "D": "derivfn"
      },
      "question": "5. A function \\( derivfn(intnum) \\) of the positive integral variable \\( intnum \\) is defined by the following properties: \\( derivfn(1)=0, derivfn(primenum)=1 \\) if \\( primenum \\) is a prime, \\( derivfn(firstvar\\,secondv)=firstvar\\,derivfn(secondv)+ secondv\\,derivfn(firstvar) \\) for any two positive integers \\( firstvar \\) and \\( secondv \\). Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely \\( \\boldsymbol{derivfn}(intnum) \\). (Derive a formula for \\( \\boldsymbol{derivfn}(intnum) / intnum \\), assuming that \\( intnum=primeone^{expoone} primetwo^{expotwo} \\cdots primeidx^{expovar} \\) where \\( primeone, primetwo, \\ldots, primeidx \\) are different primes.)\n(ii) For what values of \\( intnum \\) is \\( derivfn(intnum)=intnum \\)?\n(iii) Define \\( derivfn^{2}(intnum)=derivfn[derivfn(intnum)] \\), etc., and find the limit of \\( derivfn^{iterate}(63) \\) as \\( iterate \\) tends to \\( \\infty \\).",
      "solution": "Solution. (i) Suppose there is a function \\( derivfn \\) with the required properties. We have\n\\[\n\\frac{derivfn(firstvar\\,secondv)}{firstvar\\,secondv}= \\frac{derivfn(firstvar)}{firstvar}+ \\frac{derivfn(secondv)}{secondv}\n\\]\nand by induction\n\\[\n\\frac{derivfn\\!\\left(unitone\\,unittwo\\,\\cdots\\,unitvar\\right)}{unitone\\,unittwo\\,\\cdots\\,unitvar}= \\sum_{i} \\frac{derivfn\\!\\left(firstvar_{i}\\right)}{firstvar_{i}} .\n\\]\n\nHence\n\\[\n\\frac{derivfn\\!\\left(primenum^{exponent}\\right)}{primenum^{exponent}} = exponent\\,\\frac{derivfn(primenum)}{primenum}= \\frac{exponent}{primenum},\n\\]\nif \\( primenum \\) is a prime, and for any integer \\( intnum \\) with prime factorization \\( primenum_{1}^{exponent_{1}}\\,primenum_{2}^{exponent_{2}}\\cdots primenum_{indexer}^{exponent_{indexer}} \\) we have\n\\[\n\\frac{derivfn(intnum)}{intnum}= \\sum_{i} \\frac{exponent_{i}}{primenum_{i}} .\n\\]\n\nThis equation shows that there is at most one function with the given properties.\n\nOn the other hand, since every integer \\( intnum>1 \\) has a factorization into primes that is unique apart from order, and since the order in which the primes are numbered does not affect the sum in (2), we can define \\( derivfn(1)=0 \\) and use (2) to define \\( derivfn(intnum) \\) for \\( intnum>1 \\). So defined, \\( derivfn(primenum)=1 \\) for \\( primenum \\) a prime, and \\( derivfn \\) clearly satisfies (1), which is equivalent to \\( derivfn(firstvar\\,secondv)=firstvar\\,derivfn(secondv)+ secondv\\,derivfn(firstvar) \\). Thus there is a unique function with the prescribed properties.\n\nWe note for future reference that \\( derivfn(intnum)>0 \\) for \\( intnum>1 \\).\n\n(ii) The equation \\( derivfn(intnum)=intnum \\) is equivalent to\n\\[\n\\frac{exponent_{1}}{primenum_{1}}+\\frac{exponent_{2}}{primenum_{2}}+\\cdots+\\frac{exponent_{indexer}}{primenum_{indexer}}=1,\n\\]\nwhere \\( intnum=primenum_{1}^{exponent_{1}}\\,primenum_{2}^{exponent_{2}}\\cdots primenum_{indexer}^{exponent_{indexer}} \\) is the prime factorization of \\( intnum \\). If (3) is multiplied through by \\( primenum_{1} primenum_{2}\\cdots primenum_{indexer-1} \\), we see that \\( primenum_{1} primenum_{2}\\cdots primenum_{indexer-1}\\,exponent_{indexer}/primenum_{indexer} \\) is an integer. Since the primes are all different, we conclude that \\( primenum_{indexer} \\) divides \\( exponent_{indexer} \\). So \\( exponent_{indexer}/primenum_{indexer} \\) is an integer, and it is clear from (3) that \\( indexer=1 \\) and \\( exponent_{indexer}=primenum_{indexer} \\). Thus any solution of \\( derivfn(intnum)=intnum \\) has the form \\( intnum=primenum^{primenum} \\) where \\( primenum \\) is prime. Conversely, any such \\( intnum \\) is a solution.\n\\[\n\\begin{aligned}\n derivfn(63) & =51,\\\\\n derivfn^{2}(63) & =derivfn(51)=20,\\\\\n derivfn^{3}(63) & =derivfn(20)=24,\\\\\n derivfn^{4}(63) & =derivfn(24)=44,\\\\\n derivfn^{5}(63) & =derivfn(44)=48\n\\end{aligned}\n\\]\n\nIt appears that \\( derivfn^{iterate}(63) \\) has started to increase with \\( iterate \\).\nSuppose \\( intnum=4\\,indexer \\) where \\( indexer>1 \\). Then \\( derivfn(intnum)=derivfn(4)\\,indexer+4\\,derivfn(indexer)=4(indexer+derivfn(indexer))>4\\,indexer=intnum \\). Thus, if \\( intnum>4 \\) and \\( intnum \\) is divisible by 4, then \\( derivfn(intnum)>intnum \\) and \\( derivfn(intnum) \\) is divisible by 4. This implies that the sequence\n\\[\n derivfn(intnum),\\;derivfn^{2}(intnum),\\;derivfn^{3}(intnum),\\ldots,derivfn^{iterate}(intnum),\\ldots\n\\]\nis strictly increasing. Since \\( derivfn \\) takes integral values, \\( derivfn^{iterate}(intnum)\\to\\infty \\) as \\( iterate\\to\\infty \\) whenever \\( intnum=4\\,indexer,\\;indexer>1 \\).\n\nApplying this result to the case above, we see that \\( derivfn^{iterate}(63)=derivfn^{iterate-2}(20)\\to\\infty \\) as \\( iterate\\to\\infty \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "teacupful",
        "p": "rhinocero",
        "k": "jellyfish",
        "m": "caterpill",
        "u": "pendulums",
        "v": "diamonds",
        "u_1": "pendulumon",
        "u_2": "pendulumtw",
        "u_k": "pendulumpl",
        "p_1": "hippogriff",
        "p_2": "hippogryph",
        "p_k": "hippophony",
        "\\alpha": "semaphore",
        "\\alpha_1": "semaphorax",
        "\\alpha_2": "semaphoray",
        "\\alpha_k": "semaphoraz",
        "D": "masquerade"
      },
      "question": "5. A function \\( masquerade(teacupful) \\) of the positive integral variable \\( teacupful \\) is defined by the following properties: \\( masquerade(1)=0, masquerade(rhinocero)=1 \\) if \\( rhinocero \\) is a prime, \\( masquerade(pendulums diamonds)=pendulums masquerade(diamonds)+ \\) \\( diamonds masquerade(pendulums) \\) for any two positive integers \\( pendulums \\) and \\( diamonds \\). Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely \\( \\boldsymbol{masquerade}(teacupful) \\). (Derive a formula for \\( \\boldsymbol{masquerade}(teacupful) / teacupful \\), assuming that \\( teacupful=hippogriff^{semaphorax} hippogryph^{semaphoray} \\cdots hippophony^{semaphoraz} \\) where \\( hippogriff, hippogryph, \\ldots, hippophony \\) are different primes.)\n(ii) For what values of \\( teacupful \\) is \\( masquerade(teacupful)=teacupful \\) ?\n(iii) Define \\( masquerade^{2}(teacupful)=masquerade[masquerade(teacupful)] \\), etc., and find the limit of \\( masquerade^{caterpill}(63) \\) as \\( caterpill \\) tends to \\( \\infty \\).",
      "solution": "Solution. (i) Suppose there is a function \\( masquerade \\) with the required properties. We have\n\\[\n\\frac{masquerade(pendulums diamonds)}{pendulums diamonds}=\\frac{masquerade(pendulums)}{pendulums}+\\frac{masquerade(diamonds)}{diamonds}\n\\]\nand by induction\n\\[\n\\frac{masquerade\\left(pendulumon\\ pendulumtw\\ \\cdots\\ pendulumpl\\right)}{pendulumon\\ pendulumtw\\ \\cdots\\ pendulumpl}=\\sum_{i} \\frac{masquerade\\left(u_{i}\\right)}{u_{i}} .\n\\]\n\nHence\n\\[\n\\frac{masquerade\\left(rhinocero^{semaphore}\\right)}{rhinocero^{semaphore}}=semaphore \\frac{masquerade(rhinocero)}{rhinocero}=\\frac{semaphore}{rhinocero}\n\\]\nif \\( rhinocero \\) is a prime, and for any integer \\( teacupful \\) with prime factorization \\( hippogriff^{semaphorax} hippogryph^{semaphoray} \\cdots hippophony^{semaphoraz} \\) we have\n\\[\n\\frac{masquerade(teacupful)}{teacupful}=\\sum_{i} \\frac{\\alpha_{i}}{p_{i}}\n\\]\n\nThis equation shows that there is at most one function with the given properties.\n\nOn the other hand, since every integer \\( teacupful>1 \\) has a factorization into primes that is unique apart from order, and since the order in which the primes are numbered does not affect the sum in (2), we can define \\( masquerade(1)=0 \\) and use (2) to define \\( masquerade(teacupful) \\) for \\( teacupful>1 \\). So defined, \\( masquerade(rhinocero)=1 \\) for \\( rhinocero \\) a prime, and \\( masquerade \\) clearly satisfies (1), which is equivalent to \\( masquerade(pendulums diamonds)=pendulums masquerade(diamonds)+ diamonds masquerade(pendulums) \\). Thus there is a unique function with the prescribed properties.\n\nWe note for future reference that \\( masquerade(teacupful)>0 \\) for \\( teacupful>1 \\).\n(ii) The equation \\( masquerade(teacupful)=teacupful \\) is equivalent to\n\\[\n\\frac{semaphorax}{hippogriff}+\\frac{semaphoray}{hippogryph}+\\cdots+\\frac{semaphoraz}{hippophony}=1,\n\\]\nwhere \\( teacupful=hippogriff^{semaphorax} hippogryph^{semaphoray} \\cdots hippophony^{semaphoraz} \\) is the prime factorization of \\( teacupful \\). If (3) is multiplied through by \\( hippogriff\\ hippogryph \\cdots p_{k-1} \\), we see that \\( hippogriff\\ hippogryph \\cdots p_{k-1}\\ semaphoraz / hippophony \\) is an integer. Since the \\( p \\) 's are all different, we conclude that \\( hippophony \\) divides \\( semaphoraz \\). So \\( semaphoraz / hippophony \\) is an integer, and it is clear from (3) that \\( jellyfish=1 \\) and \\( semaphoraz=hippophony \\). Thus any solution of \\( masquerade(teacupful)=teacupful \\) has the form \\( teacupful=rhinocero^{rhinocero} \\) where \\( rhinocero \\) is prime. Conversely, any such \\( teacupful \\) is a solution.\n\\[\n\\begin{aligned}\nmasquerade(63) & =51, \\\\\nmasquerade^{2}(63) & =masquerade(51)=20, \\\\\nmasquerade^{3}(63) & =masquerade(20)=24, \\\\\nmasquerade^{4}(63) & =masquerade(24)=44, \\\\\nmasquerade^{5}(63) & =masquerade(44)=48\n\\end{aligned}\n\\]\n\nIt appears that \\( masquerade^{caterpill}(63) \\) has started to increase with \\( caterpill \\).\nSuppose \\( teacupful=4 jellyfish \\) where \\( jellyfish>1 \\). Then \\( masquerade(teacupful)=masquerade(4) jellyfish+4 masquerade(jellyfish)=4(jellyfish+ masquerade(jellyfish))>4 jellyfish=teacupful \\). Thus, if \\( teacupful>4 \\) and \\( teacupful \\) is divisible by 4 , then \\( masquerade(teacupful)>teacupful \\) and \\( masquerade(teacupful) \\) is divisible by 4 . This implies that the sequence\n\\[\nmasquerade(teacupful), masquerade^{2}(teacupful), masquerade^{3}(teacupful), \\ldots, masquerade^{caterpill}(teacupful), \\ldots\n\\]\nis strictly increasing. Since \\( masquerade \\) takes integral values, \\( masquerade^{caterpill}(teacupful) \\rightarrow \\infty \\) as \\( caterpill \\rightarrow \\infty \\) whenever \\( teacupful=4 jellyfish, jellyfish>1 \\).\n\nApplying this result to the case above, we see that \\( masquerade^{caterpill}(63)=masquerade^{caterpill-2}(20) \\)\n\\( \\rightarrow \\infty \\) as \\( caterpill \\rightarrow \\infty \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "nonpositive",
        "p": "composite",
        "k": "uncounted",
        "m": "fixedstep",
        "u": "constantval",
        "v": "staticval",
        "u_1": "constantone",
        "u_2": "constanttwo",
        "u_k": "constantunk",
        "p_1": "compositeone",
        "p_2": "compositetwo",
        "p_k": "compositeunk",
        "\\alpha": "logarithm",
        "\\alpha_1": "logarithmone",
        "\\alpha_2": "logarithmtwo",
        "\\alpha_k": "logarithmunk",
        "D": "frozenmap"
      },
      "question": "5. A function \\( frozenmap(nonpositive) \\) of the positive integral variable \\( nonpositive \\) is defined by the following properties: \\( frozenmap(1)=0, frozenmap(composite)=1 \\) if \\( composite \\) is a prime, \\( frozenmap(constantval staticval)=constantval frozenmap(staticval)+ \\) \\( staticval frozenmap(constantval) \\) for any two positive integers \\( constantval \\) and \\( staticval \\). Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely \\( \\boldsymbol{frozenmap}(nonpositive) \\). (Derive a formula for \\( \\boldsymbol{frozenmap}(nonpositive) / nonpositive \\), assuming that \\( nonpositive=compositeone^{logarithmone} compositetwo^{logarithmtwo} \\cdots compositeunk^{logarithmunk} \\) where \\( compositeone, compositetwo, \\ldots, compositeunk \\) are different primes.)\n(ii) For what values of \\( nonpositive \\) is \\( frozenmap(nonpositive)=nonpositive \\) ?\n(iii) Define \\( frozenmap^{2}(nonpositive)=frozenmap[frozenmap(nonpositive)] \\), etc., and find the limit of \\( frozenmap^{fixedstep}(63) \\) as \\( fixedstep \\) tends to \\( \\infty \\).",
      "solution": "Solution. (i) Suppose there is a function \\( frozenmap \\) with the required properties. We have\n\\[\n\\frac{frozenmap(constantval staticval)}{constantval staticval}=\\frac{frozenmap(constantval)}{constantval}+\\frac{frozenmap(staticval)}{staticval}\n\\]\nand by induction\n\\[\n\\frac{frozenmap\\left(constantone constanttwo \\cdots constantunk\\right)}{constantone constanttwo \\cdots constantunk}=\\sum_{i} \\frac{frozenmap\\left(u_{i}\\right)}{u_{i}} .\n\\]\n\nHence\n\\[\n\\frac{frozenmap\\left(composite^{logarithm}\\right)}{composite^{logarithm}}=logarithm \\frac{frozenmap(composite)}{composite}=\\frac{logarithm}{composite}\n\\]\nif \\( composite \\) is a prime, and for any integer \\( nonpositive \\) with prime factorization \\( compositeone^{logarithmone} compositetwo^{logarithmtwo} \\cdots compositeunk^{logarithmunk} \\) we have\n\\[\n\\frac{frozenmap(nonpositive)}{nonpositive}=\\sum_{i} \\frac{\\alpha_{i}}{p_{i}}\n\\]\n\nThis equation shows that there is at most one function with the given properties.\n\nOn the other hand, since every integer \\( nonpositive>1 \\) has a factorization into primes that is unique apart from order, and since the order in which the primes are numbered does not affect the sum in (2), we can define \\( frozenmap(1) \\) \\( =0 \\) and use (2) to define \\( frozenmap(nonpositive) \\) for \\( nonpositive>1 \\). So defined, \\( frozenmap(composite)=1 \\) for \\( composite \\) a prime, and \\( frozenmap \\) clearly satisfies (1), which is equivalent to \\( frozenmap(constantval staticval)=constantval frozenmap(staticval)+ \\) \\( staticval frozenmap(constantval) \\). Thus there is a unique function with the prescribed properties.\n\nWe note for future reference that \\( frozenmap(nonpositive)>0 \\) for \\( nonpositive>1 \\).\n(ii) The equation \\( frozenmap(nonpositive)=nonpositive \\) is equivalent to\n\\[\n\\frac{\\alpha_{1}}{p_{1}}+\\frac{\\alpha_{2}}{p_{2}}+\\cdots+\\frac{\\alpha_{k}}{p_{k}}=1,\n\\]\nwhere \\( nonpositive=compositeone^{logarithmone} compositetwo^{logarithmtwo} \\cdots compositeunk^{logarithmunk} \\) is the prime factorization of \\( nonpositive \\). If (3) is multiplied through by \\( compositeone compositetwo \\cdots p_{k-1} \\), we see that \\( compositeone compositetwo \\cdots p_{k-1} logarithmunk / compositeunk \\) is an integer. Since the \\( composite \\)'s are all different, we conclude that \\( compositeunk \\) divides \\( logarithmunk \\). So \\( logarithmunk / compositeunk \\) is an integer, and it is clear from (3) that \\( uncounted=1 \\) and \\( logarithmunk=compositeunk \\). Thus any solution of \\( frozenmap(nonpositive)=nonpositive \\) has the form \\( nonpositive=composite^{composite} \\) where \\( composite \\) is prime. Conversely, any such \\( nonpositive \\) is a solution.\n\\[\n\\begin{aligned}\nfrozenmap(63) & =51, \\\\\nfrozenmap^{2}(63) & =frozenmap(51)=20, \\\\\nfrozenmap^{3}(63) & =frozenmap(20)=24, \\\\\nfrozenmap^{4}(63) & =frozenmap(24)=44, \\\\\nfrozenmap^{5}(63) & =frozenmap(44)=48\n\\end{aligned}\n\\]\n\nIt appears that \\( frozenmap^{fixedstep}(63) \\) has started to increase with \\( fixedstep \\).\nSuppose \\( nonpositive=4 uncounted \\) where \\( uncounted>1 \\). Then \\( frozenmap(nonpositive)=frozenmap(4) uncounted+4 frozenmap(uncounted)=4(uncounted+ \\) \\( frozenmap(uncounted))>4 uncounted=nonpositive \\). Thus, if \\( nonpositive>4 \\) and \\( nonpositive \\) is divisible by 4 , then \\( frozenmap(nonpositive)>nonpositive \\) and \\( frozenmap(nonpositive) \\) is divisible by 4 . This implies that the sequence\n\\[\nfrozenmap(nonpositive), frozenmap^{2}(nonpositive), frozenmap^{3}(nonpositive), \\ldots, frozenmap^{fixedstep}(nonpositive), \\ldots\n\\]\nis strictly increasing. Since \\( frozenmap \\) takes integral values, \\( frozenmap^{fixedstep}(nonpositive) \\rightarrow \\infty \\) as \\( fixedstep \\rightarrow \\infty \\) whenever \\( nonpositive=4 uncounted, uncounted>1 \\).\n\nApplying this result to the case above, we see that \\( frozenmap^{fixedstep}(63)=frozenmap^{fixedstep-2}(20) \\)\n\\( \\rightarrow \\infty \\) as \\( fixedstep \\rightarrow \\infty \\)."
    },
    "garbled_string": {
      "map": {
        "n": "jkdhslae",
        "p": "qzxwvtnp",
        "k": "hjgrksla",
        "m": "vbnczxpt",
        "u": "rtyuioqw",
        "v": "poilkjmh",
        "u_1": "zmxnvbqw",
        "u_2": "trewqlas",
        "u_k": "asdfrtgh",
        "p_1": "uiopghjk",
        "p_2": "lkjhgfdw",
        "p_k": "qazwsxed",
        "\\alpha": "mnbvcxza",
        "\\alpha_1": "poiuytre",
        "\\alpha_2": "lkjasdfh",
        "\\alpha_k": "zxcvbnml",
        "D": "qweasdzx"
      },
      "question": "5. A function \\( qweasdzx(jkdhslae) \\) of the positive integral variable \\( jkdhslae \\) is defined by the following properties: \\( qweasdzx(1)=0, qweasdzx(qzxwvtnp)=1 \\) if \\( qzxwvtnp \\) is a prime, \\( qweasdzx(rtyuioqw poilkjmh)=rtyuioqw qweasdzx(poilkjmh)+ \\) \\( poilkjmh qweasdzx(rtyuioqw) \\) for any two positive integers \\( rtyuioqw \\) and \\( poilkjmh \\). Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely \\( \\boldsymbol{qweasdzx}(jkdhslae) \\). (Derive a formula for \\( \\boldsymbol{qweasdzx}(jkdhslae) / jkdhslae \\), assuming that \\( jkdhslae=uiopghjk^{poiuytre} lkjhgfdw^{lkjasdfh} \\cdots qazwsxed^{zxcvbnml} \\) where \\( uiopghjk, lkjhgfdw, \\ldots, qazwsxed \\) are different primes.)\n(ii) For what values of \\( jkdhslae \\) is \\( qweasdzx(jkdhslae)=jkdhslae \\) ?\n(iii) Define \\( qweasdzx^{2}(jkdhslae)=qweasdzx[qweasdzx(jkdhslae)] \\), etc., and find the limit of \\( qweasdzx^{vbnczxpt}(63) \\) as \\( vbnczxpt \\) tends to \\( \\infty \\).",
      "solution": "Solution. (i) Suppose there is a function \\( qweasdzx \\) with the required properties. We have\n\\[\n\\frac{qweasdzx(rtyuioqw poilkjmh)}{rtyuioqw poilkjmh}=\\frac{qweasdzx(rtyuioqw)}{rtyuioqw}+\\frac{qweasdzx(poilkjmh)}{poilkjmh}\n\\]\nand by induction\n\\[\n\\frac{qweasdzx\\left(zmxnvbqw trewqlas \\cdots asdfrtgh\\right)}{zmxnvbqw trewqlas \\cdots asdfrtgh}=\\sum_{i} \\frac{qweasdzx\\left(u_{i}\\right)}{u_{i}} .\n\\]\n\nHence\n\\[\n\\frac{qweasdzx\\left(qzxwvtnp^{mnbvcxza}\\right)}{qzxwvtnp^{mnbvcxza}}=mnbvcxza \\frac{qweasdzx(qzxwvtnp)}{qzxwvtnp}=\\frac{mnbvcxza}{qzxwvtnp}\n\\]\nif \\( qzxwvtnp \\) is a prime, and for any integer \\( jkdhslae \\) with prime factorization \\( uiopghjk^{poiuytre} lkjhgfdw^{lkjasdfh} \\cdots qazwsxed^{zxcvbnml} \\) we have\n\\[\n\\frac{qweasdzx(jkdhslae)}{jkdhslae}=\\sum_{i} \\frac{\\alpha_{i}}{p_{i}}\n\\]\n\nThis equation shows that there is at most one function with the given properties.\n\nOn the other hand, since every integer \\( jkdhslae>1 \\) has a factorization into primes that is unique apart from order, and since the order in which the primes are numbered does not affect the sum in (2), we can define \\( qweasdzx(1)=0 \\) and use (2) to define \\( qweasdzx(jkdhslae) \\) for \\( jkdhslae>1 \\). So defined, \\( qweasdzx(qzxwvtnp)=1 \\) for \\( qzxwvtnp \\) a prime, and \\( qweasdzx \\) clearly satisfies (1), which is equivalent to \\( qweasdzx(rtyuioqw poilkjmh)=rtyuioqw qweasdzx(poilkjmh)+ poilkjmh qweasdzx(rtyuioqw) \\). Thus there is a unique function with the prescribed properties.\n\nWe note for future reference that \\( qweasdzx(jkdhslae)>0 \\) for \\( jkdhslae>1 \\).\n(ii) The equation \\( qweasdzx(jkdhslae)=jkdhslae \\) is equivalent to\n\\[\n\\frac{poiuytre}{uiopghjk}+\\frac{lkjasdfh}{lkjhgfdw}+\\cdots+\\frac{zxcvbnml}{qazwsxed}=1,\n\\]\nwhere \\( jkdhslae=uiopghjk^{poiuytre} lkjhgfdw^{lkjasdfh} \\cdots qazwsxed^{zxcvbnml} \\) is the prime factorization of \\( jkdhslae \\). If (3) is multiplied through by \\( p_{1} p_{2} \\cdots p_{k-1} \\), we see that \\( p_{1} p_{2} \\cdots p_{k-1} zxcvbnml / qazwsxed \\) is an integer. Since the \\( p \\) 's are all different, we conclude that \\( qazwsxed \\) divides \\( zxcvbnml \\). So \\( zxcvbnml / qazwsxed \\) is an integer, and it is clear from (3) that \\( hjgrksla=1 \\) and \\( zxcvbnml=qazwsxed \\). Thus any solution of \\( qweasdzx(jkdhslae)=jkdhslae \\) has the form \\( jkdhslae=qzxwvtnp^{qzxwvtnp} \\) where \\( qzxwvtnp \\) is prime. Conversely, any such \\( jkdhslae \\) is a solution.\n\\[\n\\begin{aligned}\nqweasdzx(63) & =51, \\\\\nqweasdzx^{2}(63) & =qweasdzx(51)=20, \\\\\nqweasdzx^{3}(63) & =qweasdzx(20)=24, \\\\\nqweasdzx^{4}(63) & =qweasdzx(24)=44, \\\\\nqweasdzx^{5}(63) & =qweasdzx(44)=48\n\\end{aligned}\n\\]\n\nIt appears that \\( qweasdzx^{vbnczxpt}(63) \\) has started to increase with \\( vbnczxpt \\).\nSuppose \\( jkdhslae=4 hjgrksla \\) where \\( hjgrksla>1 \\). Then \\( qweasdzx(jkdhslae)=qweasdzx(4) hjgrksla+4 qweasdzx(hjgrksla)=4(hjgrksla+ qweasdzx(hjgrksla))>4 hjgrksla=jkdhslae \\). Thus, if \\( jkdhslae>4 \\) and \\( jkdhslae \\) is divisible by 4 , then \\( qweasdzx(jkdhslae)>jkdhslae \\) and \\( qweasdzx(jkdhslae) \\) is divisible by 4 . This implies that the sequence\n\\[\nqweasdzx(jkdhslae), qweasdzx^{2}(jkdhslae), qweasdzx^{3}(jkdhslae), \\ldots, qweasdzx^{vbnczxpt}(jkdhslae), \\ldots\n\\]\nis strictly increasing. Since \\( qweasdzx \\) takes integral values, \\( qweasdzx^{vbnczxpt}(jkdhslae) \\rightarrow \\infty \\) as \\( vbnczxpt \\rightarrow \\infty \\) whenever \\( jkdhslae=4 hjgrksla, hjgrksla>1 \\).\n\nApplying this result to the case above, we see that \\( qweasdzx^{vbnczxpt}(63)=qweasdzx^{vbnczxpt-2}(20) \\rightarrow \\infty \\) as \\( vbnczxpt \\rightarrow \\infty \\)."
    },
    "kernel_variant": {
      "question": "Let the arithmetic function d: ℕ → ℕ ∪ {0} be defined by\n  • d(1) = 0 ,\n  • d(p) = 1 for every prime p ,\n  • d(uv) = u d(v) + v d(u)   for all positive integers u , v .\n\nAnswer the following.\n\n(i)  Prove that the three rules above are compatible and obtain an explicit closed formula for d(n) in terms of the prime–power factorisation of n.\n\n(ii) Determine all positive integers n that satisfy the functional equation d(n) = n.\n\n(iii) For m ≥ 1 write d^{m} for the m-fold iterate of d.  Evaluate\n        lim_{m→∞} d^{m}(81).",
      "solution": "Part (i).  Compatibility and a closed formula.\n------------------------------------------------\nIntroduce the auxiliary function\n        E(n) := d(n)/n     (n ≥ 1).\nDividing the product rule by uv gives\n        E(uv) = E(u) + E(v)          (1)\nfor all positive integers u, v; that is, E is additive on the multiplicative semigroup ℕ.\n\nStep 1.  Prime powers.  Put u = p and v = p^{α-1} (α ≥ 2) in (1):\n        E(p^{α}) = E(p) + E(p^{α-1}).\nBecause E(p) = d(p)/p = 1/p, an induction on α yields\n        E(p^{α}) = α/p,          d(p^{α}) = α p^{α-1}.      (2)\n\nStep 2.  General n.  Let n = ∏_{i=1}^{k} p_i^{α_i} be the canonical prime decomposition.  Repeated use of (1) gives\n        E(n) = ∑_{i=1}^{k} E(p_i^{α_i}).\nInsert (2):\n        E(n) = ∑_{i=1}^{k} α_i / p_i.                   (3)\nHence\n        d(n) = n · ∑_{i=1}^{k} α_i / p_i.               (4)\n\nFormula (4) clearly reproduces the initial data d(1)=0, d(p)=1 and satisfies the product rule because (1) does, so the three axioms are compatible and determine d uniquely.\n\nPart (ii).  Solving d(n) = n.\n----------------------------------\nWith n = ∏ p_i^{α_i}, condition d(n) = n is equivalent, by (3), to\n        ∑_{i=1}^{k} α_i / p_i = 1.                     (5)\nMultiply (5) by P := ∏_{i=1}^{k} p_i.  Fix r and reduce mod p_r:\n        α_r · P / p_r ≡ 0 (mod p_r).\nSince gcd(P / p_r , p_r) = 1, we must have p_r | α_r, whence α_r ≥ p_r for every r.  Consequently\n        1 = ∑ α_i / p_i ≥ k,\nso k = 1 and α_1 / p_1 = 1, i.e. α_1 = p_1.  Conversely, (4) gives d(p^{p}) = p^{p}.  Therefore\n        d(n) = n   ⇔   n = p^{p} with p prime.\n\nPart (iii).  The iterates of d at 81.\n----------------------------------------\nWe first prove two lemmas.\n\nLemma 1.  If 27 divides n, then 27 divides d(n).\nProof.  Write n = 27k.  Using the product rule and (2) with α = 3,\n        d(n) = d(27k) = 27 d(k) + k d(27) = 27 d(k) + k · 27 = 27 (d(k) + k),\nso 27 | d(n). ∎\n\nLemma 2.  If 27 divides n and n > 27, then d(n) > n.  Moreover d(27) = 27.\nProof.  Express n as n = 3^{β} m with β ≥ 3 and 3 ∤ m.  From (3),\n        d(n)/n = β/3 + Σ_{p|m} α_p / p ≥ β/3 ≥ 1,\nand the inequality is strict because β ≥ 3 and n > 27 forces β ≥ 4 or m > 1, either of which makes the right–hand side strictly larger than 1.  Thus d(n) > n whenever n > 27 and 27 | n.  Direct calculation from (2) gives d(27) = 27. ∎\n\nNow analyse the forward orbit of 81:\nInitial value: 81 = 3^{4} > 27, so 27 | 81.\nInductive step:  If 27 | n and n > 27, Lemma 1 gives 27 | d(n) and Lemma 2 gives d(n) > n.  Thus the sequence\n        81, d(81), d^{2}(81), …\nremains divisible by 27 and is strictly increasing.  Being an increasing sequence of integers, it must diverge to +∞.  Therefore\n        lim_{m→∞} d^{m}(81) = ∞.\n\n(For completeness the first few values are\n   d(81)  = 108,\n   d^{2}(81) = 216,\n   d^{3}(81) = 540,\n   d^{4}(81) = 1188,\n   d^{5}(81) = 2592, …,  increasing without bound.)\n\nHence the limit in part (iii) is +∞.",
      "_meta": {
        "core_steps": [
          "Rewrite the rule as D(uv)/(uv)=D(u)/u + D(v)/v and extend it inductively to any product.",
          "Apply it to prime powers to get D(n)/n = Σ α_i / p_i, which also proves existence–uniqueness of D.",
          "Set D(n)=n, obtaining Σ α_i / p_i =1; this forces n to be a single prime power with exponent equal to that prime (n = p^p).",
          "Use D(p^p k)=p^p(k + D(k)) > p^p k for k>1 to show that every multiple of p^p (bigger than p^p) grows strictly under iteration and stays a multiple of p^p.",
          "Iterate the given starting number until a multiple of p^p (>p^p) appears; thereafter the sequence diverges to infinity, giving the required limit."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Choice of the prime p whose self–power p^p is used as the ‘expanding’ modulus (the text picks p=2, giving 4). Any prime would work the same way.",
            "original": "p = 2  (so p^p = 4)"
          },
          "slot2": {
            "description": "Initial argument whose trajectory is examined in part (iii). It only needs the property that some iterate becomes a multiple of p^p larger than p^p.",
            "original": "63"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}