is prime iff the 14 Diophantine equations in 26 variables
have a solution in positive integers (Jones et al. 1976; Riesel 1994, p. 40).
More things to try:
Weisstein, Eric W. "Prime Diophantine Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeDiophantineEquations.html