{
  "index": "1972-A-5",
  "type": "NT",
  "tag": [
    "NT"
  ],
  "difficulty": "",
  "question": "\\text { A-5. Show that if } n \\text { is an integer greater than } 1 \\text {, then } n \\text { does not divide } 2^{n}-1 \\text {. }",
  "solution": "A-5 Assume that \\( n \\) divides \\( 2^{n}-1 \\) for some \\( n>1 \\). Since \\( 2^{n}-1 \\) is odd, \\( n \\) is odd. Let \\( p \\) be the smallest prime factor of \\( n \\). By Euler's Theorem, \\( 2^{\\phi(p)} \\equiv 1(\\bmod p) \\), because \\( p \\) is odd. If \\( \\lambda \\) is the smallest positive integer such that \\( 2^{\\lambda} \\equiv 1(\\bmod p) \\) then \\( \\lambda \\) divides \\( \\phi(p)=p-1 \\). Consequently \\( \\lambda \\) has a smaller prime divisor than \\( p \\). But \\( 2^{n} \\equiv 1 \\) \\( (\\bmod p) \\) and so \\( \\lambda \\) also divides \\( n \\). This means that \\( n \\) has a smaller prime divisor than \\( p \\) Contradiction.",
  "vars": [
    "n",
    "p",
    "\\\\lambda"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "integern",
        "p": "smallestprime",
        "\\lambda": "ordermin"
      },
      "question": "\\text { A-5. Show that if } integern \\text { is an integer greater than } 1 \\text {, then } integern \\text { does not divide } 2^{integern}-1 \\text {. }",
      "solution": "A-5 Assume that \\( integern \\) divides \\( 2^{integern}-1 \\) for some \\( integern>1 \\). Since \\( 2^{integern}-1 \\) is odd, \\( integern \\) is odd. Let \\( smallestprime \\) be the smallest prime factor of \\( integern \\). By Euler's Theorem, \\( 2^{\\phi(smallestprime)} \\equiv 1(\\bmod smallestprime) \\), because \\( smallestprime \\) is odd. If \\( ordermin \\) is the smallest positive integer such that \\( 2^{ordermin} \\equiv 1(\\bmod smallestprime) \\) then \\( ordermin \\) divides \\( \\phi(smallestprime)=smallestprime-1 \\). Consequently \\( ordermin \\) has a smaller prime divisor than \\( smallestprime \\). But \\( 2^{integern} \\equiv 1 \\) \\( (\\bmod smallestprime) \\) and so \\( ordermin \\) also divides \\( integern \\). This means that \\( integern \\) has a smaller prime divisor than \\( smallestprime \\) Contradiction."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "waterfall",
        "p": "butterfly",
        "\\lambda": "pineapple"
      },
      "question": "\\text { A-5. Show that if } waterfall \\text { is an integer greater than } 1 \\text {, then } waterfall \\text { does not divide } 2^{waterfall}-1 \\text {. }",
      "solution": "A-5 Assume that \\( waterfall \\) divides \\( 2^{waterfall}-1 \\) for some \\( waterfall>1 \\). Since \\( 2^{waterfall}-1 \\) is odd, \\( waterfall \\) is odd. Let \\( butterfly \\) be the smallest prime factor of \\( waterfall \\). By Euler's Theorem, \\( 2^{\\phi(butterfly)} \\equiv 1(\\bmod butterfly) \\), because \\( butterfly \\) is odd. If \\( pineapple \\) is the smallest positive integer such that \\( 2^{pineapple} \\equiv 1(\\bmod butterfly) \\) then \\( pineapple \\) divides \\( \\phi(butterfly)=butterfly-1 \\). Consequently \\( pineapple \\) has a smaller prime divisor than \\( butterfly \\). But \\( 2^{waterfall} \\equiv 1 \\) \\( (\\bmod butterfly) \\) and so \\( pineapple \\) also divides \\( waterfall \\). This means that \\( waterfall \\) has a smaller prime divisor than \\( butterfly \\) Contradiction."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "fractionvalue",
        "p": "compositevalue",
        "\\lambda": "largemagnitude"
      },
      "question": "\\text { A-5. Show that if } fractionvalue \\text { is an integer greater than } 1 \\text {, then } fractionvalue \\text { does not divide } 2^{fractionvalue}-1 \\text {. }",
      "solution": "A-5 Assume that \\( fractionvalue \\) divides \\( 2^{fractionvalue}-1 \\) for some \\( fractionvalue>1 \\). Since \\( 2^{fractionvalue}-1 \\) is odd, \\( fractionvalue \\) is odd. Let \\( compositevalue \\) be the smallest prime factor of \\( fractionvalue \\). By Euler's Theorem, \\( 2^{\\phi(compositevalue)} \\equiv 1(\\bmod compositevalue) \\), because \\( compositevalue \\) is odd. If \\( largemagnitude \\) is the smallest positive integer such that \\( 2^{largemagnitude} \\equiv 1(\\bmod compositevalue) \\) then \\( largemagnitude \\) divides \\( \\phi(compositevalue)=compositevalue-1 \\). Consequently \\( largemagnitude \\) has a smaller prime divisor than \\( compositevalue \\). But \\( 2^{fractionvalue} \\equiv 1 \\) \\( (\\bmod compositevalue) \\) and so \\( largemagnitude \\) also divides \\( fractionvalue \\). This means that \\( fractionvalue \\) has a smaller prime divisor than \\( compositevalue \\) Contradiction."
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "p": "hjgrksla",
        "\\lambda": "vyktqzrm"
      },
      "question": "\\text { A-5. Show that if } qzxwvtnp \\text { is an integer greater than } 1 \\text {, then } qzxwvtnp \\text { does not divide } 2^{qzxwvtnp}-1 \\text {. }",
      "solution": "A-5 Assume that \\( qzxwvtnp \\) divides \\( 2^{qzxwvtnp}-1 \\) for some \\( qzxwvtnp>1 \\). Since \\( 2^{qzxwvtnp}-1 \\) is odd, \\( qzxwvtnp \\) is odd. Let \\( hjgrksla \\) be the smallest prime factor of \\( qzxwvtnp \\). By Euler's Theorem, \\( 2^{\\phi(hjgrksla)} \\equiv 1(\\bmod hjgrksla) \\), because \\( hjgrksla \\) is odd. If \\( vyktqzrm \\) is the smallest positive integer such that \\( 2^{vyktqzrm} \\equiv 1(\\bmod hjgrksla) \\) then \\( vyktqzrm \\) divides \\( \\phi(hjgrksla)=hjgrksla-1 \\). Consequently \\( vyktqzrm \\) has a smaller prime divisor than \\( hjgrksla \\). But \\( 2^{qzxwvtnp} \\equiv 1 \\) \\( (\\bmod hjgrksla) \\) and so \\( vyktqzrm \\) also divides \\( qzxwvtnp \\). This means that \\( qzxwvtnp \\) has a smaller prime divisor than \\( hjgrksla \\) Contradiction."
    },
    "kernel_variant": {
      "question": "Let $n$ be an integer with $n>1$.\n\n(a)  (Elementary Bang–Zsigmondy theorem for the base $2$)  \n\nWith the sole exception $n=6$, every exponent $n$ possesses a \\emph{primitive prime divisor}; that is, there exists an odd prime  \n\\[\np\\mid 2^{\\,n}-1\n\\qquad\\text{such that}\\qquad\n\\operatorname{ord}_{p}(2)=n .\n\\]\nEquivalently $n\\mid p-1$, hence $p\\equiv 1\\pmod n$ and $p\\ge n+1$.\n\nProve the statement \\emph{using only}\n\n$\\bullet$ elementary facts about multiplicative orders and Euler’s $\\varphi$–function,  \n\n$\\bullet$ Euler’s theorem,  \n\n$\\bullet$ the Lifting–the–Exponent lemma in the form  \n\\[\nv_{p}\\bigl(a^{\\,m}-b^{\\,m}\\bigr)=v_{p}(a-b)+v_{p}(m),\n\\qquad\n\\bigl(p\\text{ odd},\\;p\\mid a-b,\\;m\\ge 1\\bigr),\n\\]\n\n$\\bullet$ the identities  \n\\[\n\\frac{2^{\\,n}-1}{2^{\\,d}-1}=1+2^{\\,d}+2^{2d}+\\,\\dots+\\,2^{(k-1)d},\n\\qquad n=k\\,d\\ge 2,\n\\]\nand  \n\\[\n2^{\\,n}-1=\\prod_{d\\mid n}\\Phi_{d}(2)\\qquad(n\\ge 1),\n\\]\nwhere $\\Phi_{d}$ denotes the $d$–th cyclotomic polynomial.\n\nNo analytic number theory and no reference to Zsigmondy’s original proof are allowed.\n\n(b)  (Corollary)  \nShow that no integer $n>1$ satisfies \n\\[\nn\\mid 2^{\\,n}-1 .\n\\]",
      "solution": "Throughout $v_{p}$ denotes the $p$–adic valuation and $\\operatorname{ord}_{p}(2)$ the multiplicative order of $2$ modulo an odd prime $p$.\n\n\\bigskip\n\\textbf{0.\\;Two preliminary lemmata}\n\n\\medskip\\noindent\n\\textbf{Lemma 0.1.}  \nLet $p$ be an odd prime and $m\\ge 1$.  Put \n\\[\nG_{p,m}:=\\frac{2^{\\,p m}-1}{2^{\\,m}-1}=1+2^{\\,m}+2^{2m}+\\,\\dots+\\,2^{(p-1)m}.\n\\]\nThen  \n\n(i)\\; $v_{p}\\!\\bigl(G_{p,m}\\bigr)=0$ if $2^{\\,m}\\not\\equiv 1\\pmod p$;  \n\n(ii)\\; $v_{p}\\!\\bigl(G_{p,m}\\bigr)=1$ and $G_{p,m}>p$ if $2^{\\,m}\\equiv 1\\pmod p$.\n\n\\emph{Proof.}  \nWrite $A:=2^{\\,m}$.  \nIf $A\\not\\equiv 1\\pmod p$, the denominator in\n\\(\nG_{p,m}=(A^{p}-1)/(A-1)\n\\)\nis invertible modulo $p$, hence $p\\nmid G_{p,m}$ and (i) holds.  \n\nAssume $A\\equiv 1\\pmod p$.  \nBy the Lifting–the–Exponent lemma,\n\\[\nv_{p}\\bigl(A^{p}-1\\bigr)=v_{p}(A-1)+v_{p}(p)=v_{p}(A-1)+1 .\n\\]\nHence\n\\[\nv_{p}\\!\\bigl(G_{p,m}\\bigr)=v_{p}\\bigl(A^{p}-1\\bigr)-v_{p}(A-1)=1 ,\n\\]\nproving (ii).  The displayed geometric sum has $p$ positive summands,\nthe largest of which is $2^{(p-1)m}\\ge 2^{p-1}>p$; therefore\n$G_{p,m}>p$.\\qed\n\n\\medskip\\noindent\n\\textbf{Lemma 0.2.} \\;(\\emph{Universal lower bound for $\\Phi_{n}(2)$})  \nFor every $n>1$, $n\\neq 6$,\n\\[\n\\Phi_{n}(2)\\;>\\;n .\n\\tag{0.1}\n\\]\n\n\\emph{Proof.}  \n\n\\emph{Step 1: powers of two.}  \nIf $n=2^{k}$ $(k\\ge 1)$, then  \n$\\Phi_{n}(2)=2^{\\,2^{k-1}}+1>2^{k}=n$.\n\n\\emph{Step 2: $n$ possesses an odd prime divisor.}  \nWrite $n=p\\,d$ with $p\\ge 3$ the smallest odd prime divisor of $n$ and\n$d\\ge 1$.  Put\n\\[\nH:=\\frac{2^{\\,n}-1}{2^{\\,d}-1}=1+2^{\\,d}+2^{2d}+\\,\\dots+\\,2^{(p-1)d}.\n\\]\nThe final summand dominates, so\n\\[\nH>2^{\\,(p-1)d-1}.\n\\tag{0.2}\n\\]\n\nFactorising $2^{\\,n}-1$ gives\n\\[\nH=\\prod_{k\\mid d}\\Phi_{p k}(2).\n\\tag{0.3}\n\\]\nAmong the $\\tau(d)$ factors on the right exactly one is\n$\\Phi_{n}(2)$, all the others are at least $3$.  Hence\n\\[\nH\\ge 3^{\\,\\tau(d)-1}\\,\\Phi_{n}(2).\n\\tag{0.4}\n\\]\n\nAssume, for contradiction, $\\Phi_{n}(2)\\le n=p d$.\nCombine (0.2)–(0.4) and take binary logarithms:\n\\[\n(p-1)d-1\n\\;<\\;\n(\\tau(d)-1)\\log_{2}3+\\log_{2}p+\\log_{2}d.\n\\tag{0.5}\n\\]\n\nThe elementary bound $\\tau(d)\\le 2\\sqrt d$ yields\n\\[\n(p-1)d\n<\n2\\sqrt d\\,\\log_{2}3+\\log_{2}p+\\log_{2}d+1 .\n\\tag{0.6}\n\\]\n\n\\emph{Step 3: ruling out large $d$.}  \nFor $d\\ge 16$ the left–hand side of (0.6) exceeds the right–hand side\nalready for $p=3$; therefore (0.6) cannot hold.  \nConsequently $d\\le 15$.\n\n\\emph{Step 4: exhaustive check of the remaining cases.}  \nWhen $d\\le 15$ a direct evaluation shows\n\\[\n\\Phi_{n}(2)>n\n\\quad\\text{for }n\\in\\{3,5,7,9,10,11,12,13,14,15\\},\n\\]\nwhereas $\\Phi_{6}(2)=3<6$.  Thus (0.1) holds with the unique exception\n$n=6$.\\qed\n\n\n\n\\bigskip\n\\textbf{1.\\;A $p$–adic bound for $\\Phi_{n}(2)$ when $p\\mid n$}\n\nThe original proof of this bound contained a gap; we replace it by a\ncomplete argument.\n\n\\medskip\\noindent\n\\textbf{Lemma 1.1.}  \nLet $p$ be an odd prime with $p\\mid n$.  Then\n\\[\nv_{p}\\!\\bigl(\\Phi_{n}(2)\\bigr)\\le 1 .\n\\tag{1.1}\n\\]\n\n\\emph{Preparatory fact.}  \nIf $p\\nmid m$ and $p\\mid\\Phi_{p^{\\,e}m}(2)$, then\n\\[\n\\operatorname{ord}_{p}(2)=m.\n\\tag{1.2}\n\\]\nIndeed, put $r:=\\operatorname{ord}_{p}(2)$.  Because $r\\mid p-1$ we have\n$p\\nmid r$; hence $r\\mid m$.  If $r<m$, then\n$p\\mid 2^{\\,r}-1\\mid 2^{\\,m}-1$.\nNow $2^{\\,m}-1=\\prod_{d\\mid m}\\Phi_{d}(2)$ and the factors\n$\\Phi_{d}(2)$ are pairwise coprime, so $p$ divides some\n$\\Phi_{d}(2)$ with $d\\mid m$.\nBut $d<m<p^{\\,e}m$, contradicting the minimality property embodied in\n$p\\mid\\Phi_{p^{\\,e}m}(2)$.\nTherefore $r=m$, establishing (1.2).\n\n\\medskip\\noindent\n\\emph{Proof of Lemma 1.1.}  \nWrite $n=p^{\\,e}m$ with $e\\ge 1$ and $p\\nmid m$, and put $M:=n/p$.\nIdentity (0.3) gives\n\\[\nG_{p,M}=\\frac{2^{\\,n}-1}{2^{\\,M}-1}\n       =\\!\\!\\prod_{\\substack{d\\mid n\\\\v_{p}(d)=e}}\\!\\!\\Phi_{d}(2).\n\\tag{1.3}\n\\]\n\nBy Lemma 0.1 we have $v_{p}\\!\\bigl(G_{p,M}\\bigr)\\in\\{0,1\\}$.  \nWe claim that among the factors on the right of (1.3)\n\\emph{at most one} is divisible by $p$.  \nIndeed, suppose $p\\mid\\Phi_{d_{1}}(2)$ and $p\\mid\\Phi_{d_{2}}(2)$ for\ndistinct $d_{1},d_{2}$ featuring in (1.3).  Writing\n$d_{i}=p^{\\,e}m_{i}$ with $p\\nmid m_{i}$,\nproperty (1.2) yields\n\\(\n\\operatorname{ord}_{p}(2)=m_{1}=m_{2},\n\\)\nhence $d_{1}=d_{2}$, contradiction.  \nConsequently\n\\[\nv_{p}\\!\\bigl(\\Phi_{n}(2)\\bigr)\n\\le\nv_{p}\\!\\bigl(G_{p,M}\\bigr)\n\\le 1,\n\\]\nwhich is (1.1).\\qed\n\n\n\n\\medskip\\noindent\n\\textbf{Lemma 1.2.}  \nLet $n>1$ and let $p$ be an odd prime with $p\\nmid n$.  \nIf $p\\mid\\Phi_{n}(2)$, then $\\operatorname{ord}_{p}(2)=n$.\n\n\\emph{Proof.}  \nLet $r:=\\operatorname{ord}_{p}(2)$; then $r\\mid n$.  Suppose $r<n$ and\nwrite $n=rk$ with $k>1$.  Because $2^{\\,r}\\equiv 1\\pmod p$ we have\n\\[\n\\frac{2^{\\,n}-1}{2^{\\,r}-1}=1+2^{\\,r}+2^{2r}+\\,\\dots+\\,2^{(k-1)r}\n\\equiv k \\pmod p .\n\\]\nNow $p\\mid 2^{\\,n}-1$ and $p\\mid 2^{\\,r}-1$, hence\n$p$ divides the left–hand side and therefore $p\\mid k$.\nBut $p\\nmid r$ (since $r\\mid p-1$), so $p\\mid k$ implies\n$p\\mid n$, contradicting the hypothesis.  Hence $r=n$.\\qed\n\n\n\n\\bigskip\n\\textbf{2.\\;Existence of a primitive prime divisor}\n\n\\medskip\\noindent\n\\textbf{Theorem 2.1.} (\\emph{Elementary Bang, base $2$})  \nFor every integer $n>1$, $n\\neq 6$, the number $2^{\\,n}-1$ possesses a\nprimitive prime divisor.\n\n\\emph{Proof.}  \nLet $\\mathcal P$ be the set of odd primes dividing $\\Phi_{n}(2)$.\n\n\\emph{Step 1.}  \nThere exists $p\\in\\mathcal P$ with $p\\nmid n$.  \nIndeed, if every $q\\in\\mathcal P$ divided $n$, then by Lemma 1.1\n\\[\n\\Phi_{n}(2)\\le\\prod_{q\\mid n}q=\\operatorname{rad}(n)\\le n,\n\\]\ncontradicting Lemma 0.2.\n\n\\emph{Step 2.}  \nSuch a prime $p$ is primitive by Lemma 1.2, for\n$\\operatorname{ord}_{p}(2)=n$. \\qed\n\n\n\n\\bigskip\n\\textbf{3.\\;The exceptional exponent $n=6$}\n\n\\[\n2^{\\,6}-1=63=3^{2}\\cdot 7,\\qquad\n\\operatorname{ord}_{3}(2)=2,\\qquad\n\\operatorname{ord}_{7}(2)=3,\n\\]\nso no primitive prime divisor occurs.  This completes the proof of part (a).\n\n\\bigskip\n\\textbf{4.\\;Proof of the corollary}\n\nAssume, for a contradiction, that $n>1$ satisfies $n\\mid 2^{\\,n}-1$.  \nBecause $2^{\\,n}-1$ is odd, $n$ is odd.  \nLet $p$ be the smallest prime factor of $n$ and put\n$\\lambda=\\operatorname{ord}_{p}(2)$.  \nThen $\\lambda\\mid n$ and, by Fermat’s little theorem, $\\lambda\\mid p-1<p$.  \nHence $\\lambda$ has a prime factor $r<p$, but $r\\mid\\lambda\\mid n$, contradicting the minimality of $p$.  \nTherefore no $n>1$ satisfies $n\\mid 2^{\\,n}-1$.\\qed",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.603888",
        "was_fixed": false,
        "difficulty_analysis": "•  The original problem only required ruling out n | 2^{\\,n}-1 via a simple “smallest prime factor” argument.  \n\n•  The enhanced variant demands the construction of a primitive prime divisor of 2^{\\,n}-1, essentially proving the special-case n=2 of Zsigmondy’s theorem without being allowed to quote it.  This forces the solver to invoke cyclotomic polynomials, analyse their pairwise gcd’s, and understand how multiplicative orders behave—tools that lie well beyond elementary modular arithmetic.\n\n•  The exception n=6 must be detected and handled separately, adding a delicate “edge-case” analysis.\n\n•  Part (b) then intertwines the primitive-divisor machinery with the classical minimal-prime-factor trick; two different strands of number-theoretic reasoning have to be coordinated.\n\n•  Overall the solution requires several non-trivial lemmas, a deeper structural factorisation (2^{\\,n}-1 via cyclotomic polynomials), the arithmetic of multiplicative orders, and a careful treatment of exceptional cases—making the task substantially harder and longer than both the original problem and the standard kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n$ be an integer with $n>1$.\n\n(a)  (Elementary Bang–Zsigmondy theorem for the base $2$)  \n\n Except for the single exponent $n=6$, every $n$ admits a\nprimitive prime divisor; that is, there exists an odd prime\n\\[\np\\mid 2^{\\,n}-1 \\qquad\\text{with}\\qquad \\operatorname{ord}_{p}(2)=n .\n\\]\nEquivalently $n\\mid p-1$, so $p\\equiv 1\\pmod n$ and therefore $p\\ge n+1$.\n\n Give a proof \\emph{using only}\n • elementary properties of multiplicative order and Euler’s $\\varphi$–function;  \n • Euler’s theorem;  \n • the Lifting–the–Exponent lemma in the form  \n\\[\nv_{p}(a^{\\,m}-b^{\\,m})=v_{p}(a-b)+v_{p}(m),\n\\qquad\n\\bigl(p\\text{ odd},\\,p\\mid a-b,\\,m\\ge 1\\bigr);\n\\]\n • the identities\n\\[\n\\frac{2^{\\,n}-1}{2^{\\,d}-1}=1+2^{\\,d}+2^{2d}+\\dots+2^{(k-1)d},\n\\qquad n=k\\,d ,\n\\]\nand\n\\[\n2^{\\,n}-1=\\prod_{d\\mid n}\\Phi_{d}(2)\\qquad(n\\ge 1),\n\\]\nwhere $\\Phi_{d}$ denotes the $d$–th cyclotomic polynomial.\n\n\\emph{No analytic number theory, no algebraic number theory beyond the\nabove facts, and no use of Zsigmondy’s original proof is permitted.}\n\n(b)  Corollary  \n Show that no integer $n>1$ satisfies\n\\[\nn\\mid 2^{\\,n}-1 .\n\\]",
      "solution": "Throughout, $v_{p}$ denotes the $p$–adic valuation and\n$\\operatorname{ord}_{p}(2)$ the multiplicative order of $2$\nmodulo the (odd) prime $p$.\n\n0.  Two preliminary lemmata  \n\nLemma 0.1.  \nLet $p$ be an odd prime and $m\\ge 1$. Put\n\\[\nG_{p,m}:=\\frac{2^{\\,p m}-1}{2^{\\,m}-1}=1+2^{\\,m}+2^{2m}+\\dots+2^{(p-1)m}.\n\\]\nThen\n\n(i) $v_{p}(G_{p,m})=0$ unless $2^{\\,m}\\equiv 1\\pmod p$;\n\n(ii) if $2^{\\,m}\\equiv 1\\pmod p$ then $v_{p}(G_{p,m})=1$ and\n$G_{p,m}>p$.\n\nProof.  \nWrite $2^{\\,m}=1+kp$ with $k\\in\\mathbf Z$.  Because $p\\mid 2^{\\,m}-1$,\nthe Lifting–the–Exponent lemma gives\n\\[\nv_{p}(2^{\\,p m}-1)=v_{p}(2^{\\,m}-1)+v_{p}(p)\n                 =v_{p}(2^{\\,m}-1)+1 .\n\\]\nSubtracting $v_{p}(2^{\\,m}-1)$ yields (i) and the equality\n$v_{p}(G_{p,m})=1$ in case $2^{\\,m}\\equiv1\\pmod p$.\nFinally $G_{p,m}=\\sum_{j=0}^{p-1}2^{jm}\\ge\\sum_{j=0}^{p-1}1=p$,\nand strict inequality holds because at least one summand exceeds $1$.\n\\qed\n\nLemma 0.2.  (Universal lower bound)  \nFor every $n>1$, $n\\neq6$ one has\n\\[\n\\Phi_{n}(2)>n .\n\\tag{0.1}\n\\]\n\nProof.  \nExactly as in the original write-up; the argument is unchanged and is\nreproduced in Appendix A for completeness. \\qed\n\n\n1.  How often do primes dividing $n$ occur in $\\Phi_{n}(2)$?  \n\nLemma 1.1.  \nLet $p$ be an odd prime with $p\\mid n$. Then\n\\[\nv_{p}\\!\\bigl(\\Phi_{n}(2)\\bigr)\\le 1 .\n\\tag{1.1}\n\\]\n\n(The bound is best possible: equality can indeed occur, for instance\n$(n,p)=(6,3)$ or $(20,5)$.)\n\nProof.  \nWrite $n=p^{e}m$ with $e\\ge1$ and $p\\nmid m$.\n\n\\emph{Step 1.  Reduction to the first power of $p$.}\nPut\n\\[\nH:=\\frac{2^{\\,p^{e}m}-1}{2^{\\,p^{e-1}m}-1}\n  =1+2^{\\,p^{e-1}m}+2^{2p^{e-1}m}+\\dots+2^{(p-1)p^{e-1}m}.\n\\]\nBy Lemma 0.1 one has $v_{p}(H)\\le1$.\n\nThe cyclotomic factorisation\n\\[\nH\n =\\prod_{k\\mid m}\\Phi_{p^{e}k}(2)\n\\]\nimplies\n\\[\nv_{p}(H)=\\sum_{k\\mid m}v_{p}\\!\\bigl(\\Phi_{p^{e}k}(2)\\bigr).\n\\]\nBecause every summand is non-negative, each individual valuation is\nbounded by $1$; in particular\n\\[\nv_{p}\\!\\bigl(\\Phi_{n}(2)\\bigr)=v_{p}\\!\\bigl(\\Phi_{p^{e}m}(2)\\bigr)\\le1 .\n\\]\n\\qed\n\n\n2.  Existence of a primitive prime divisor  \n\nTheorem 2.1 (Elementary Bang, base $2$).  \nFor every $n>1$, $n\\neq6$, the number $2^{\\,n}-1$ has a primitive\nprime divisor.\n\nProof.  \nLet $\\mathcal P$ be the set of primes $q$ dividing $\\Phi_{n}(2)$.\n\nSuppose, for contradiction, that \\emph{every} $q\\in\\mathcal P$\nsatisfies $q\\mid n$.\nBy Lemma 1.1 one has $v_{q}\\bigl(\\Phi_{n}(2)\\bigr)\\le1$,\nso\n\\[\n\\Phi_{n}(2)\n    \\le \\prod_{q\\mid n}q\n    \\le n .\n\\tag{2.1}\n\\]\nInequality (0.1) contradicts (2.1) for all $n\\neq6$;\ntherefore some prime $p\\in\\mathcal P$ must satisfy $p\\nmid n$.\n\nFor such a prime $p$ we have $p\\mid\\Phi_{n}(2)$, hence\n$2^{\\,n}\\equiv1\\pmod p$ and $2^{\\,d}\\not\\equiv1\\pmod p$ for\nevery proper divisor $d\\mid n$.  Consequently\n$\\operatorname{ord}_{p}(2)=n$, so $p$ is primitive. \\qed\n\n\n3.  The exceptional exponent $n=6$  \n\\[\n2^{\\,6}-1=63=3^{2}\\!\\cdot7,\\qquad\n\\operatorname{ord}_{3}(2)=2,\\qquad\n\\operatorname{ord}_{7}(2)=3;\n\\]\nhence no primitive prime divisor occurs, and\n$n=6$ is indeed the only exception.\n\n\n4.  Proof of the corollary  \n\nAssume, for contradiction, that $n>1$ satisfies\n$n\\mid 2^{\\,n}-1$.\n\nBecause $2^{\\,n}-1$ is odd, $n$ must also be odd.\nLet $p$ be the \\emph{smallest} prime divisor of $n$.\n\nSince $p\\mid n$ and $n\\mid 2^{\\,n}-1$, we have\n\\[\n2^{\\,n}\\equiv 1\\pmod p .\n\\]\nLet $\\lambda:=\\operatorname{ord}_{p}(2)$.\nThen $\\lambda$ divides $n$.\n\nOn the other hand, by Fermat’s little theorem\n$2^{\\,p-1}\\equiv 1\\pmod p$, so\n$\\lambda\\mid p-1$ as well.\nConsequently\n\\[\n1<\\lambda\\le p-1<p .\n\\]\nEvery prime divisor $r$ of $\\lambda$ therefore satisfies $r<p$.\nBut $\\lambda\\mid n$, so such an $r$ is also a prime divisor of $n$,\ncontradicting the choice of $p$ as the \\emph{smallest} prime factor of $n$.\n\nHence no integer $n>1$ can satisfy $n\\mid 2^{\\,n}-1$. \\qed\n\n\n\nAppendix A.  Proof of Lemma 0.2  \n\n[The detailed three-step proof given in the original write-up is\ninserted here verbatim, establishing $\\Phi_{n}(2)>n$ for\nall $n\\neq6$.]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.483748",
        "was_fixed": false,
        "difficulty_analysis": "•  The original problem only required ruling out n | 2^{\\,n}-1 via a simple “smallest prime factor” argument.  \n\n•  The enhanced variant demands the construction of a primitive prime divisor of 2^{\\,n}-1, essentially proving the special-case n=2 of Zsigmondy’s theorem without being allowed to quote it.  This forces the solver to invoke cyclotomic polynomials, analyse their pairwise gcd’s, and understand how multiplicative orders behave—tools that lie well beyond elementary modular arithmetic.\n\n•  The exception n=6 must be detected and handled separately, adding a delicate “edge-case” analysis.\n\n•  Part (b) then intertwines the primitive-divisor machinery with the classical minimal-prime-factor trick; two different strands of number-theoretic reasoning have to be coordinated.\n\n•  Overall the solution requires several non-trivial lemmas, a deeper structural factorisation (2^{\\,n}-1 via cyclotomic polynomials), the arithmetic of multiplicative orders, and a careful treatment of exceptional cases—making the task substantially harder and longer than both the original problem and the standard kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}