We are heading out for a little vacation before school starts and I wanted a gentle topic for today’s project. When I woke up this morning the idea of introducing the boys to quadratic reciprocity jumped into my head. The Wikipedia page on the topic gave me a few ideas:
I started the project by showing them the chart on Wikipedia’s page showing the factorization of for integers 1, 2, 3, and etc . . .
What patterns, if any, would they see?
Next we moved to a second table from Wikipedia’s page – this table shows the squares mod p for primes going going from 3 to 47.
Again, what patterns, if any, do they notice?
Now I had them look for a special number -> for which primes p could we find a square congruent to -1 mod p?
Finally, we wrote short program in Mathematica to test the conjecture that we had in the last video. The conjecture was that primes congruent to 3 mod 4 would have no squares congruent to -1 mod p, and for primes congruent to 1 mod 4 would, -1 would always be a square.
Sorry for the less than stellar camera work in this video . . . .