Today I was looking for a project with my son and flipped open to the chapter on quadratic reciprocity. It had a few introductory ideas that I thought would be fun to share with my younger son.

We first looked at Wilson’s Theorem:

After Wilson’s theorem, we moved on to talking about perfect squares mod a prime. After a fairly long discussion here my son noticed that half of the non-zero number mod a prime are perfect squares:

Finally, I asked him to make a mod 11 multiplication table and we talked through some of the patterns in the table – including that the non-zero numbers had multiplicative inverses:

It was a really fun discussion today. I know next to nothing about number theory, but I really would like to use Weissman’s book more to explore some advanced ideas with the boys.

One thought on “More intro number theory with my son inspired by Martin Weissman’s An Illustrated Theory of Numbers”

hi
you can now prove example forms of wilson theorem. for example using the 11 table you had, you could ask your son to write the inverse of 1, 2, 3 and so forth. You can notice then that if the inverse of , say, 2 is 6 then the inverse of 6 is 2, and check this with some further examples. You can also notice the unique numbers having inverses equal to themselves are both 1 and 10 (or p-1). For every other number its inverse will equal some other number. Now write 1.2.3.4.5.6.7.8.9.10.
every number appearing in this expression has also its inverse there, except for 1 and 10. So you can use 2.6=1 , 3.4=1 , 5.9=1, 7.8=1, to kill all the terms but 1 and 10. Hence the result.

hi

you can now prove example forms of wilson theorem. for example using the 11 table you had, you could ask your son to write the inverse of 1, 2, 3 and so forth. You can notice then that if the inverse of , say, 2 is 6 then the inverse of 6 is 2, and check this with some further examples. You can also notice the unique numbers having inverses equal to themselves are both 1 and 10 (or p-1). For every other number its inverse will equal some other number. Now write 1.2.3.4.5.6.7.8.9.10.

every number appearing in this expression has also its inverse there, except for 1 and 10. So you can use 2.6=1 , 3.4=1 , 5.9=1, 7.8=1, to kill all the terms but 1 and 10. Hence the result.