Diophantine Equation

A Diophantine equation is a polynomial equation with integer coefficients for which only integer solutions are sought.

BBdio(H) is the maximum score among all solvable polynomial Diophantine equations of height H. To determine the height of an equation, write it in the form P(x,y,z,...) = 0, then take the sum of |coefficient|*2^degree among all terms in the LHS polynomial P. To determine the score of a solvable equation, find the solution with the lowest score, where the score of a solution is the maximum absolute value among all variables.

BBdio(n) is proven for n < 16.

BBdio(n)	Value	Champion
2 ≤ n ≤ 20	≥ n-2
21	≥ 26	$x^2+xy+6y+1=0$
22	≥ 22	$x^3+y^2+4=3x$
23	≥ 31	$x^2+xy+x+1=6y$
24	≥ 66	$x^3+xy+5y=2$
25	≥ 470	$x^3+xy+y^2+y+7=0$
26	≥ 40	$x^2+xy+7y+4=0$
27	≥ 849	$x^3+x^2+y^2+9=x$
28	≥ 74	$x^3+xy+2x+5y=2$
29	≥ 221	$x^3+xy+x+7y+1=0$
30	≥ 849	$x^3+2x^2=y^2+10$