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Exponential Diophantine equations

In an exponential Diophantine equation the unknown appears in the exponent, so the equation is not a polynomial at all — \(3^m - 2^n = 1\), \(2^n - 7 = x^2\). These are exactly the equations that sympy's diophantine cannot express (see the performance page), and they are decided by some of the deepest theorems in the subject.

In[1]:= Solve[3^m - 2^n == 1 && m > 0 && n > 0, {m, n}, Integers]
Out[1]= {{m -> 1, n -> 1}, {m -> 2, n -> 3}}

The only powers of \(3\) and \(2\) that differ by \(1\) are \(3^1 - 2^1 = 1\) and \(3^2 - 2^3 = 9 - 8 = 1\).

1. Catalan's equation and Mihăilescu's theorem

Catalan conjectured in 1844 that \(8\) and \(9\) are the only consecutive perfect powers — that \(x^a - y^b = 1\) with \(x, y, a, b > 1\) has the unique solution \(3^2 - 2^3 = 1\). This resisted proof for 158 years until Preda Mihăilescu settled it in 2002 using the theory of cyclotomic fields. Mathilda knows the theorem:

In[1]:= Solve[x^a - y^b == 1 && x > 1 && y > 1 && a > 1 && b > 1, {x, y, a, b}, Integers]
Out[1]= {{x -> 3, y -> 2, a -> 2, b -> 3}}

The single tuple \((x, y, a, b) = (3, 2, 2, 3)\) is the entire solution set of the Catalan equation — one of the rare Diophantine problems whose answer is a theorem with a name. The same machinery handles fixed-base variants like the \(3^m - 2^n = 1\) shown above, where Mihăilescu's result plus the small linear cases give the complete list.

References

  1. E. Catalan, Note extraite d'une lettre adressée à l'éditeur, J. Reine Angew. Math. 27 (1844), 192.
  2. P. Mihăilescu, Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math. 572 (2004), 167–195.

2. The Ramanujan–Nagell equation \(2^n - 7 = x^2\)

In 1913 Ramanujan asked which \(2^n - 7\) are perfect squares; Nagell proved in 1948 that there are exactly five. It is the prototypical equation of the shape \(x^2 + D = 2^n\). Mathilda solves it by factoring in the ring \(\mathbb{Z}[(1 + \sqrt{-7})/2]\) (which has class number one), reducing the problem to a Lucas-sequence term equal to \(\pm 1\); the Bilu–Hanrot–Voutier primitive-divisor theorem then bounds \(n \le 32\), so a short finite scan returns the whole set:

In[1]:= Solve[2^n - 7 == x^2 && n > 0 && x > 0, {n, x}, Integers]
Out[1]= {{n -> 3, x -> 1}, {n -> 4, x -> 3}, {n -> 5, x -> 5}, {n -> 7, x -> 11}, {n -> 15, x -> 181}}

The five solutions — the largest being \(2^{15} - 7 = 32761 = 181^2\) — are the famous Ramanujan–Nagell numbers. The \(n = 15\) case is the reason the problem is celebrated: it is far enough out that no short search would think to look there, yet the primitive-divisor bound proves nothing lies beyond it.

Outside the class-number-one gate the general analysis (linear forms in logarithms for an arbitrary base or \(D\)) is not built, so such inputs are declined rather than half-answered:

In[2]:= Solve[3^n - 7 == x^2 && n > 0 && x > 0, {n, x}, Integers]
Out[2]= Solve[-7 + 3^n == x^2 && n > 0 && x > 0, {n, x}, Integers]

The equation returns unchanged — a reported gap, not a wrong or empty answer.

References

  1. S. Ramanujan, Question 464, J. Indian Math. Soc. 5 (1913), 130.
  2. T. Nagell, The Diophantine equation \(x^2 + 7 = 2^n\), Ark. Mat. 4 (1961), 185–187.
  3. Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122.

Next: Famous results and impossibility proofs — the showpieces the whole engine was built to reach: the Lander–Parkin and Frye counterexamples, taxicab numbers, the Euler brick, and Fermat's Last Theorem.