This paper is concerned with the ( p 1 , p 2 , p 3 )-generation of finite groups of Lie type, where we say that a group is ( p 1 , p 2 , p 3 )-generated if it is generated by two elements of orders p 1 , p 2 having product of order p 3 . Given a triple ( p 1 , p 2 , p 3 ) of primes, we say that ( p 1 , p 2 , p 3 ) is rigid for a simple algebraic group G if the sum of the dimensions of the subvarieties of elements of orders dividing p 1 , p 2 , p 3 in G is equal to 2 dim G . We conjecture that if ( p 1 , p 2 , p 3 ) is a rigid triple for G then given a prime p , there are only finitely many positive integers r such that the finite group G ( p r ) is a ( p 1 , p 2 , p 3 )-group. We prove that the conjecture holds in many cases. Finally, we classify the rigid triples for simple algebraic groups. The conjecture together with this classification puts into context many results on Hurwitz (2, 3, 7)-generation in the literature, and motivates a new study of the ( p 1 , p 2 , p 3 )-generation problem for certain finite groups of Lie type of low rank.