An ant resides at the origin of the Cartesian plane. One morning she sets out on a long excursion of its first quadrant and pledges to walk a different prime number of units every day starting with 2, always in a straight line, always within the same quadrant, and always staying overnight at lattice points. Each day, the number of units she walks is the prime number immediately following the prime number of the previous day (though in a different direction so as not to walk in a straight line on two consecutive days). At the end of her excursion, she returns home never having crossed or retraced her own path.
(Her walk will thus be a polygon of sides -in order- 2, 3, 5, 7, ..., with one of its vertices in the origin and all the others in lattice points in the first quadrant.)
a) Strictly following these rules, what is the earliest the ant can be back at the origin?
b) Can she make such excursions of any number of days greater than that minimum?
An earlier related puzzle: An ant's walk in the Cartesian Plane