Values of a quadratic expression with powers of 2 and 3

Determine whether infinitely many integers t_0 fail to admit integers m,n such that t_0+a2^m+b3^n is a square, for fixed integers a and b; more generally, establish an analogous result when 2^m and 3^n are replaced by arbitrary S-units.

Background

The paper applies its general theorem on value sets of rational functions on semi-abelian varieties to elliptic curves and multiplicative groups. It notes that replacing the elliptic curve by the projective line leads to difficult questions concerning values of expressions involving powers of 2 and 3.

The specific unresolved issue asks whether there are infinitely many integers t_0 for which the Diophantine equation t_0+a2m+b3n=x2 has no solution in integers m,n,x. The authors also state that they do not see a route to proving the analogous assertion for more general S-units without assuming strong Diophantine conjectures.

References

We note that changing E with P1 would generally lead to problems which to our knowledge are difficult and widely open. For instance, the case f (x, z1, z2) = x2 −az1 −bz2 leads to the issue of understanding whether for ‘many’ integers t0 there are no integers m, n such that t0 + a2m + b3n is a square.

— Rational and integral values of rational functions at rational points  (2608.28255 - Corvaja et al., 28 Aug 2026) in Example 1.3, page 4