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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:
- 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 + 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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