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Sage doesn't work that way, e.g. k0 in ZZ and j in ZZ evaluate to False because they are symbolic variables. You meant assume(k0, 'integer') and assume(j, 'integer'). Still, solve doesn't seem very good at your problem. Instead, you can define your solution set as a Polyhedron and ask for its integral points:

sage: p = 5
sage: j = 1
sage: Polyhedron(ieqs=[[0,1,0], [p-2,-1,0]], eqns=[[-k, 1,p-1]]).integral_points()
((1, 0),)

Sage doesn't work that way, e.g. k0 in ZZ and j in ZZ evaluate to False because they are symbolic variables. You meant assume(k0, 'integer') and assume(j, 'integer'). Still, solve doesn't seem very good at your problem. Instead, you can define your solution set as a Polyhedron and ask for its integral points:

sage: p = 5
sage: j k = 1
sage: Polyhedron(ieqs=[[0,1,0], [p-2,-1,0]], eqns=[[-k, 1,p-1]]).integral_points()
((1, 0),)