At the recent CNTA conference, Professor Joe Silverman gave an explicit homomorphism of into which he jokingly called a great question to ask undergraduates to work out explicitly… if you don’t like them very much. In a similar vein, here is a question that one might ask undergraduates they don’t particularly like:
Let be a triple of co-prime integers which are not all zero and such that . Prove that the ternary quadratic form given by:
takes on integer values whenever the triplet lies in the lattice defined by the congruence conditions
I will reveal the solution in due time (and it does not involve explicit computation of congruences), but if come up with a solution let me know!