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We study the class of non-holonomic power series with integer coefficientsthat reduce, modulo primes, or powers of primes, to algebraic functions. Inparticular we try to determine whether the susceptibility of the square-latticeIsing model belongs to this class, and more broadly whether the susceptibilityis a solution of a differentially algebraic equation. Initial results onTutte's non-linear ordinary differential equation (ODE) and other simplequadratic non-linear ODEs suggest that a large set of differentially algebraicpower series solutions with integer coefficients might reduce to algebraicfunctions modulo primes, or powers of primes. Here we give several examples ofseries with integer coefficients and non-zero radius of convergence that reduceto algebraic functions modulo (almost) every prime (or power of a prime). Theseexamples satisfy differentially algebraic equations with the encodingpolynomial occasionally possessing quite high degree (and thus difficult toidentify even with long series). Additionally, we have extended both the high-and low-temperature Ising square-lattice susceptibility series to 5043coefficients. We find that even this long series is insufficient to determinewhether it reduces to algebraic functions modulo $3$, $5$, etc. This negativeresult is in contrast to the comparatively easy confirmation that thecorresponding series reduce to algebraic functions modulo powers of $2$.

Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?