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2011 Paper 1 Q8
D: 1516.0 B: 1484.0
proof integer solutions Diophantine equation cubic inequality number theory squeezing argument

  1. The numbers \(m\) and \(n\) satisfy \[ m^3=n^3+n^2+1\,. \tag{\(*\)} \]
    • Show that \(m > n\). Show also that \(m < n+1\) if and only if \(2n^2+3n > 0\,\). Deduce that \(n < m < n+1\) unless \(-\frac32 \le n \le 0\,\).
    • Hence show that the only solutions of \((*)\) for which both \(m\) and \(n\) are integers are \((m,n) = (1,0)\) and \((m,n)= (1,-1)\).
  2. Find all integer solutions of the equation \[ p^3=q^3+2q^2-1\,. \]

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