Bowen’s morning class notes: July 2, 2002
by Art Mabbott
No multiples of 11, yet.
If p is prime, and p ≠ a^{2} + b^{2},
then no multiples of p are in the list until p^{2}.
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 5^{3}).
Closed under multiplication.
If n = a^{2} + b^{2}, and m = c^{2} + d^{2},
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 = a^{2} + b^{2}
Problem:
 Find all of the numbers less than 100 that can be expressed as the sum of squares of two integers.
See table above.
 Find at least three more examples
(a+bi)(c+di) = ac + bci + adi + bdi^{2} = (ac  bd) + (ad + bc)i.
Norm of a complex number (a + bi) is N(a + bi) = a^{2} + b^{2}
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 +cx^{2}+dx^{3}+…
221 = 13 17 = (3^{2}+2^{2})(4^{2}+1^{2}) = (2^{2}+3^{2})(4^{2}+1^{2})
= (3^{2}+2^{2})(1^{2}+4^{2}) =(2^{2}+3^{2})(1^{2}+4^{2})
= 10^{2}+ 11^{2} (3 + 2i)(4 + 1i) = 10 + 11i
= (5)^{2}+14^{2 } (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
