Finicky Diophantine Equations

As was observed elsewhere, Diophantine equations are quite finicky: slight modifications may lead from a solvable equation to one with no solutions. Methods that work for one equation may prove useless for a very similar equation. (To remind, Diophantine equations are polynomial equations with integer coefficients in one or more variables. The solutions sought are also integers. Here I would like to give another example of that phenomenon.

Start with the equation

x³ + y³ + z³ = 29.

This equation has an almost obvious solution (3, 1, 1) and a less obvious one (4, -3, -2). So, in the very least it is solvable. Now, change the equation to

x³ + y³ + z³ = 30.

Just to save you the effort, the solution with the smallest magnitudes is (2220422932, -2218888517, -283059965). Next modification leads to an equation with no solutions:

x³ + y³ + z³ = 31.

To see that no integer x, y, z may satisfy that equation, note that modulo 9 all integer cubes are equal to ±1 or 0. However 31≡4 (mod 9). However, 4 and 5 are the two residues modulo 9 that could not be represented as a sum of the residues ±1 and 0. So next equation

x³ + y³ + z³ = 32

is also unsolvable. Now, add 1 one more time:

x³ + y³ + z³ = 33.

Whether that equation has or has not solution is an open question. As Michael Stoll, has observed, if you were able to solve this, you should consider making Diophantine equations your research area.

References

  1. Michael Stoll, From Sex to Quadratic Forms, An Invitation to Mathematics, D. Schleicher, M. Lackmann (eds), Springer, 2011

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