Bowen’s morning class notes: July 2, 2002
by Art Mabbott

No multiples of 11, yet.
If p is prime, and p ≠ a2 + b2, then no multiples of p are in the list until p2.
Primes p = 3n + 3 don’t show up in the list.
Any number x = 4n + 3 doesn’t show on the list.
If a number can be written in more than one way, then it’s a multiple of 5. (up to 53).
Closed under multiplication.
If n = a2 + b2, and m = c2 + d2, then nm = (   )2 + (   )2
Powers of 2 are all in the list.
Even numbers of the form 2k, k = 4n + 3, don’t show up.
If P = 4n + 1 then p = a2 + b2

Problem:

  1. Find all of the numbers less than 100 that can be expressed as the sum of squares of two integers.

    See table above.

  2. Find at least three more examples

(a+bi)(c+di) = ac + bci + adi + bdi2 = (ac - bd) + (ad + bc)i.  
Norm of a complex number (a + bi) is N(a + bi) = a2 + b2
N(Z) = Z Zbar

A  Gaussian integer is a complex number of the form   a + bi where a and b are integers.

Z(i) = Gassian integers
Z(x) = a+bx +cx2+dx3+…

221     = 13 17 = (32+22)(42+12) = (22+32)(42+12) = (32+22)(12+42) =(22+32)(12+42)
          = 102+ 112                                        (3 + 2i)(4 + 1i) = 10 + 11i
          = (-5)2+142                                       (3 + 2i)(4 + 1i) = -5 + 14i

Gaussian Integers

Choose any three gaussian integers.
             (2+i), (2+3i), (3+i)
Norm       5         13        10    = 650

                        (2+i)(2+3i)(3+i) = -5+25i       Norm = 650
but                   (2+i)(3+2i)(3+i) = -5+25i       Norm = 650
and                  (1+2i)(2+3i)(3+i) = -19+17i   Norm = 650

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